Theorem btwnconn1lem13 32206

Description: Lemma for btwnconn1 32208. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.)

Assertion

Ref

Expression

btwnconn1lem13

|- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) ) -> ( C = c \/ D = d ) )

Proof of Theorem btwnconn1lem13

Dummy variables e p q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ne 2795	. . 3 |- ( C =/= c <-> -. C = c )
2		simp2rl 1130	. . . . . . . . . 10 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> C Btwn <. A , d >. )
3	2	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> C Btwn <. A , d >. )
4		simp2ll 1128	. . . . . . . . . 10 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> D Btwn <. A , c >. )
5	4	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> D Btwn <. A , c >. )
6	3, 5	jca 554	. . . . . . . 8 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( C Btwn <. A , d >. /\ D Btwn <. A , c >. ) )
7		simpl1 1064	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> N e. NN )
8		simprl1 1106	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> C e. ( EE ` N ) )
9		simpl2 1065	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> A e. ( EE ` N ) )
10		simprrl 804	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> d e. ( EE ` N ) )
11		btwncom 32121	. . . . . . . . . 10 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( C Btwn <. A , d >. <-> C Btwn <. d , A >. ) )
12	7, 8, 9, 10, 11	syl13anc 1328	. . . . . . . . 9 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( C Btwn <. A , d >. <-> C Btwn <. d , A >. ) )
13		simprl2 1107	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> D e. ( EE ` N ) )
14		simprl3 1108	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> c e. ( EE ` N ) )
15		btwncom 32121	. . . . . . . . . 10 |- ( ( N e. NN /\ ( D e. ( EE ` N ) /\ A e. ( EE ` N ) /\ c e. ( EE ` N ) ) ) -> ( D Btwn <. A , c >. <-> D Btwn <. c , A >. ) )
16	7, 13, 9, 14, 15	syl13anc 1328	. . . . . . . . 9 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( D Btwn <. A , c >. <-> D Btwn <. c , A >. ) )
17	12, 16	anbi12d 747	. . . . . . . 8 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( ( C Btwn <. A , d >. /\ D Btwn <. A , c >. ) <-> ( C Btwn <. d , A >. /\ D Btwn <. c , A >. ) ) )
18	6, 17	syl5ib 234	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( C Btwn <. d , A >. /\ D Btwn <. c , A >. ) ) )
19		axpasch 25821	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. ( EE ` N ) /\ c e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( C Btwn <. d , A >. /\ D Btwn <. c , A >. ) -> E. e e. ( EE ` N ) ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) )
20	7, 10, 14, 9, 8, 13, 19	syl132anc 1344	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( ( C Btwn <. d , A >. /\ D Btwn <. c , A >. ) -> E. e e. ( EE ` N ) ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) )
21	18, 20	syld 47	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> E. e e. ( EE ` N ) ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) )
22	21	imp 445	. . . . 5 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) ) -> E. e e. ( EE ` N ) ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) )
23		simpll1 1100	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> N e. NN )
24	14	adantr 481	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> c e. ( EE ` N ) )
25	8	adantr 481	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> C e. ( EE ` N ) )
26	10	adantr 481	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> d e. ( EE ` N ) )
27		axsegcon 25807	. . . . . . . . . 10 |- ( ( N e. NN /\ ( c e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> E. p e. ( EE ` N ) ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) )
28	23, 24, 25, 25, 26, 27	syl122anc 1335	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> E. p e. ( EE ` N ) ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) )
29		simpr 477	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> e e. ( EE ` N ) )
30		axsegcon 25807	. . . . . . . . . 10 |- ( ( N e. NN /\ ( d e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ e e. ( EE ` N ) ) ) -> E. r e. ( EE ` N ) ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) )
31	23, 26, 25, 25, 29, 30	syl122anc 1335	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> E. r e. ( EE ` N ) ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) )
32		reeanv 3107	. . . . . . . . 9 |- ( E. p e. ( EE ` N ) E. r e. ( EE ` N ) ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) <-> ( E. p e. ( EE ` N ) ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ E. r e. ( EE ` N ) ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) )
33	28, 31, 32	sylanbrc 698	. . . . . . . 8 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> E. p e. ( EE ` N ) E. r e. ( EE ` N ) ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) )
34	33	adantr 481	. . . . . . 7 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> E. p e. ( EE ` N ) E. r e. ( EE ` N ) ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) )
35	7	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) -> N e. NN )
36		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) -> p e. ( EE ` N ) )
37		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) -> r e. ( EE ` N ) )
38		axsegcon 25807	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) /\ ( r e. ( EE ` N ) /\ p e. ( EE ` N ) ) ) -> E. q e. ( EE ` N ) ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) )
39	35, 36, 37, 37, 36, 38	syl122anc 1335	. . . . . . . . . . . 12 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) -> E. q e. ( EE ` N ) ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) )
40	39	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) ) -> E. q e. ( EE ` N ) ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) )
41		simp-4l 806	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
42		simplrl 800	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) )
43	42	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) )
44	10	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> d e. ( EE ` N ) )
45		simprrr 805	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> b e. ( EE ` N ) )
46	45	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> b e. ( EE ` N ) )
47		simpllr 799	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> e e. ( EE ` N ) )
48	44, 46, 47	3jca 1242	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ e e. ( EE ` N ) ) )
49	43, 48	jca 554	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ e e. ( EE ` N ) ) ) )
50		simplrl 800	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> p e. ( EE ` N ) )
51		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> q e. ( EE ` N ) )
52		simplrr 801	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> r e. ( EE ` N ) )
53	50, 51, 52	3jca 1242	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( p e. ( EE ` N ) /\ q e. ( EE ` N ) /\ r e. ( EE ` N ) ) )
54	41, 49, 53	3jca 1242	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ e e. ( EE ` N ) ) ) /\ ( p e. ( EE ` N ) /\ q e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) )
55		simp1ll 1124	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> A =/= B )
56	55	ad3antrrr 766	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) -> A =/= B )
57	56	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> A =/= B )
58		simp1lr 1125	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> B =/= C )
59	58	ad3antrrr 766	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) -> B =/= C )
60	59	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> B =/= C )
61		simpllr 799	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) -> C =/= c )
62	61	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> C =/= c )
63	57, 60, 62	3jca 1242	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( A =/= B /\ B =/= C /\ C =/= c ) )
64		simpl1r 1113	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) )
65	64	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) )
66	63, 65	jca 554	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) )
67		simpll2 1101	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) -> ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) )
68	67	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) )
69		simpl3l 1116	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) )
70	69	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) )
71		simpl3r 1117	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) )
72	71	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) )
73	70, 72	jca 554	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) )
74	66, 68, 73	3jca 1242	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) )
75		simpllr 799	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) )
76		simplrl 800	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) )
77		simplrr 801	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) )
78		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) )
79	76, 77, 78	3jca 1242	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) )
80	74, 75, 79	jca32 558	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) ) ) )
81		btwnconn1lem12 32205	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ e e. ( EE ` N ) ) ) /\ ( p e. ( EE ` N ) /\ q e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) ) ) ) -> D = d )
82	54, 80, 81	syl2an 494	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) /\ ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) ) -> D = d )
83	82	an4s 869	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) ) /\ ( q e. ( EE ` N ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) ) -> D = d )
84	40, 83	rexlimddv 3035	. . . . . . . . . 10 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) ) -> D = d )
85	84	an4s 869	. . . . . . . . 9 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) /\ ( ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) ) -> D = d )
86	85	exp32 631	. . . . . . . 8 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> ( ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) -> ( ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) -> D = d ) ) )
87	86	rexlimdvv 3037	. . . . . . 7 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> ( E. p e. ( EE ` N ) E. r e. ( EE ` N ) ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) -> D = d ) )
88	34, 87	mpd 15	. . . . . 6 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> D = d )
89	88	an4s 869	. . . . 5 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) ) /\ ( e e. ( EE ` N ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> D = d )
90	22, 89	rexlimddv 3035	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) ) -> D = d )
91	90	expr 643	. . 3 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) ) -> ( C =/= c -> D = d ) )
92	1, 91	syl5bir 233	. 2 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) ) -> ( -. C = c -> D = d ) )
93	92	orrd 393	1 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) ) -> ( C = c \/ D = d ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-btwn 25772  df-cgr 25773  df-ofs 32090  df-colinear 32146  df-ifs 32147  df-cgr3 32148  df-fs 32149

This theorem is referenced by:  btwnconn1lem14  32207