Theorem btwnconn1lem14 32207

Description: Lemma for btwnconn1 32208. Final statement of the theorem when B =/= C. (Contributed by Scott Fenton, 9-Oct-2013.)

Assertion

Ref	Expression
btwnconn1lem14	|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) ) -> ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) )

Proof of Theorem btwnconn1lem14

Dummy variables b c d x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> N e. NN )
2		simp2l 1087	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
3		simp3r 1090	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
4		simp3 1063	. . . . 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 ) ) )
5		axsegcon 25807	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> E. c e. ( EE ` N ) ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) )
6	1, 2, 3, 4, 5	syl121anc 1331	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> E. c e. ( EE ` N ) ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) )
7		simp3l 1089	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
8		axsegcon 25807	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> E. d e. ( EE ` N ) ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) )
9	1, 2, 7, 4, 8	syl121anc 1331	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> E. d e. ( EE ` N ) ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) )
10		reeanv 3107	. . . 4 |- ( E. c e. ( EE ` N ) E. d e. ( EE ` N ) ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) <-> ( E. c e. ( EE ` N ) ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ E. d e. ( EE ` N ) ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) )
11	6, 9, 10	sylanbrc 698	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> E. c e. ( EE ` N ) E. d e. ( EE ` N ) ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) )
12	11	adantr 481	. 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) ) -> E. c e. ( EE ` N ) E. d e. ( EE ` N ) ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) )
13		simpl1 1064	. . . . . . . . . . 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 ) ) ) -> N e. NN )
14		simpl2l 1114	. . . . . . . . . . 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 ) ) ) -> A e. ( EE ` N ) )
15		simprl 794	. . . . . . . . . . 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 ) ) ) -> c e. ( EE ` N ) )
16		simpl3l 1116	. . . . . . . . . . 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 ) ) ) -> C e. ( EE ` N ) )
17		simpl2r 1115	. . . . . . . . . . 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 ) )
18		axsegcon 25807	. . . . . . . . . . 11 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> E. b e. ( EE ` N ) ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) )
19	13, 14, 15, 16, 17, 18	syl122anc 1335	. . . . . . . . . 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 ) ) ) -> E. b e. ( EE ` N ) ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) )
20		simprr 796	. . . . . . . . . . 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 ) ) ) -> d e. ( EE ` N ) )
21		simpl3r 1117	. . . . . . . . . . 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 ) ) ) -> D e. ( EE ` N ) )
22		axsegcon 25807	. . . . . . . . . . 11 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ d e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) )
23	13, 14, 20, 21, 17, 22	syl122anc 1335	. . . . . . . . . 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 ) ) ) -> E. x e. ( EE ` N ) ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) )
24	19, 23	jca 554	. . . . . . . . 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 ) ) ) -> ( E. b e. ( EE ` N ) ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ E. x e. ( EE ` N ) ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) )
25	24	adantr 481	. . . . . . . 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 ) ) ) /\ ( ( ( 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 >. ) ) ) ) -> ( E. b e. ( EE ` N ) ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ E. x e. ( EE ` N ) ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) )
26		reeanv 3107	. . . . . . . 8 |- ( E. b e. ( EE ` N ) E. x e. ( EE ` N ) ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) <-> ( E. b e. ( EE ` N ) ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ E. x e. ( EE ` N ) ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) )
27	25, 26	sylibr 224	. . . . . . 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 ) ) ) /\ ( ( ( 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 >. ) ) ) ) -> E. b e. ( EE ` N ) E. x e. ( EE ` N ) ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) )
28	13, 14, 17	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 ) ) ) -> ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
29	28	adantr 481	. . . . . . . . . . . . 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 ) /\ x e. ( EE ` N ) ) ) -> ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
30	16, 21, 15	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 ) ) ) -> ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) )
31	30	adantr 481	. . . . . . . . . . . . 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 ) /\ x e. ( EE ` N ) ) ) -> ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) )
32		simplrr 801	. . . . . . . . . . . . . 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 ) /\ x e. ( EE ` N ) ) ) -> d e. ( EE ` N ) )
33		simprl 794	. . . . . . . . . . . . . 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 ) /\ x e. ( EE ` N ) ) ) -> b e. ( EE ` N ) )
34		simprr 796	. . . . . . . . . . . . . 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 ) /\ x e. ( EE ` N ) ) ) -> x e. ( EE ` N ) )
35	32, 33, 34	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 ) /\ x e. ( EE ` N ) ) ) -> ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ x e. ( EE ` N ) ) )
36	29, 31, 35	3jca 1242	. . . . . . . . . . . 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 ) /\ x 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 ) /\ x e. ( EE ` N ) ) ) )
37		simpll 790	. . . . . . . . . . . . 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) -> ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) )
38		simplr 792	. . . . . . . . . . . . 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) -> ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) )
39		simpr 477	. . . . . . . . . . . . 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) -> ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) )
40	37, 38, 39	3jca 1242	. . . . . . . . . . . 12 |- ( ( ( ( ( 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) -> ( ( ( 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) )
41		btwnconn1lem2 32195	. . . . . . . . . . . 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 ) /\ x 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) ) -> x = b )
42	36, 40, 41	syl2an 494	. . . . . . . . . . 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 ) /\ x 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) ) -> x = b )
43		opeq2 4403	. . . . . . . . . . . . . . . . . 18 |- ( x = b -> <. A , x >. = <. A , b >. )
44	43	breq2d 4665	. . . . . . . . . . . . . . . . 17 |- ( x = b -> ( d Btwn <. A , x >. <-> d Btwn <. A , b >. ) )
45		opeq2 4403	. . . . . . . . . . . . . . . . . 18 |- ( x = b -> <. d , x >. = <. d , b >. )
46	45	breq1d 4663	. . . . . . . . . . . . . . . . 17 |- ( x = b -> ( <. d , x >.Cgr <. D , B >. <-> <. d , b >.Cgr <. D , B >. ) )
47	44, 46	anbi12d 747	. . . . . . . . . . . . . . . 16 |- ( x = b -> ( ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) <-> ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) )
48	47	anbi2d 740	. . . . . . . . . . . . . . 15 |- ( x = b -> ( ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) <-> ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) )
49	48	anbi2d 740	. . . . . . . . . . . . . 14 |- ( x = b -> ( ( ( ( ( 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) <-> ( ( ( ( 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 >. ) ) ) ) )
50	49	biimpac 503	. . . . . . . . . . . . 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) /\ x = b ) -> ( ( ( ( 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 >. ) ) ) )
51	32, 33	jca 554	. . . . . . . . . . . . . . . 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 ) /\ x e. ( EE ` N ) ) ) -> ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) )
52	29, 31, 51	jca32 558	. . . . . . . . . . . . . . 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 ) /\ x 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 ) ) ) ) )
53		simpll 790	. . . . . . . . . . . . . . . 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 >. ) ) ) -> ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) )
54		simplr 792	. . . . . . . . . . . . . . . 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 >. ) ) ) -> ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) )
55		simpr 477	. . . . . . . . . . . . . . . 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 Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) )
56	53, 54, 55	3jca 1242	. . . . . . . . . . . . . . 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 >. ) ) ) -> ( ( ( 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 >. ) ) ) )
57		btwnconn1lem13 32206	. . . . . . . . . . . . . . 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 ) ) ) ) /\ ( ( ( 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 ) )
58	52, 56, 57	syl2an 494	. . . . . . . . . . . . . 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 ) /\ x 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 ) )
59	58	ex 450	. . . . . . . . . . . . 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 ) /\ x 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 ) ) )
60	50, 59	syl5 34	. . . . . . . . . . . 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 ) /\ x 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) /\ x = b ) -> ( C = c \/ D = d ) ) )
61	60	expdimp 453	. . . . . . . . . . 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 ) /\ x 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) ) -> ( x = b -> ( C = c \/ D = d ) ) )
62	42, 61	mpd 15	. . . . . . . . . 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 ) /\ x 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 , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) ) -> ( C = c \/ D = d ) )
63	62	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 ) ) ) /\ ( ( ( 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 >. ) ) ) ) /\ ( ( b e. ( EE ` N ) /\ x e. ( EE ` N ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) ) ) -> ( C = c \/ D = d ) )
64	63	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 ) ) ) /\ ( ( ( 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 >. ) ) ) ) -> ( ( b e. ( EE ` N ) /\ x e. ( EE ` N ) ) -> ( ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) -> ( C = c \/ D = d ) ) ) )
65	64	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 ) ) ) /\ ( ( ( 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 >. ) ) ) ) -> ( E. b e. ( EE ` N ) E. x e. ( EE ` N ) ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , x >. /\ <. d , x >.Cgr <. D , B >. ) ) -> ( C = c \/ D = d ) ) )
66	27, 65	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 ) ) ) /\ ( ( ( 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 = c \/ D = d ) )
67		orcom 402	. . . . . . 7 |- ( ( C = c \/ D = d ) <-> ( D = d \/ C = c ) )
68		simprrl 804	. . . . . . . . . 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 , d >. )
69	68	adantl 482	. . . . . . . . 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 ) ) ) /\ ( ( ( 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 , d >. )
70		opeq2 4403	. . . . . . . . . 10 |- ( D = d -> <. A , D >. = <. A , d >. )
71	70	breq2d 4665	. . . . . . . . 9 |- ( D = d -> ( C Btwn <. A , D >. <-> C Btwn <. A , d >. ) )
72	69, 71	syl5ibrcom 237	. . . . . . . 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 ) ) ) /\ ( ( ( 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 >. ) ) ) ) -> ( D = d -> C Btwn <. A , D >. ) )
73		simprll 802	. . . . . . . . . 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 >. ) ) ) -> D Btwn <. A , c >. )
74	73	adantl 482	. . . . . . . . 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 ) ) ) /\ ( ( ( 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 >. ) ) ) ) -> D Btwn <. A , c >. )
75		opeq2 4403	. . . . . . . . . 10 |- ( C = c -> <. A , C >. = <. A , c >. )
76	75	breq2d 4665	. . . . . . . . 9 |- ( C = c -> ( D Btwn <. A , C >. <-> D Btwn <. A , c >. ) )
77	74, 76	syl5ibrcom 237	. . . . . . . 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 ) ) ) /\ ( ( ( 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 = c -> D Btwn <. A , C >. ) )
78	72, 77	orim12d 883	. . . . . . 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 ) ) ) /\ ( ( ( 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 >. ) ) ) ) -> ( ( D = d \/ C = c ) -> ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) ) )
79	67, 78	syl5bi 232	. . . . . 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 ) ) ) /\ ( ( ( 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 = c \/ D = d ) -> ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) ) )
80	66, 79	mpd 15	. . . . 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 ) ) ) /\ ( ( ( 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 , D >. \/ D Btwn <. A , C >. ) )
81	80	an4s 869	. . . 4 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) ) /\ ( ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) ) ) -> ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) )
82	81	exp32 631	. . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) ) -> ( ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) -> ( ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) -> ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) ) ) )
83	82	rexlimdvv 3037	. 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) ) -> ( E. c e. ( EE ` N ) E. d e. ( EE ` N ) ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) -> ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) ) )
84	12, 83	mpd 15	1 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) ) -> ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) )

Syntax hints:   -> wi 4   \/ 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:  btwnconn1  32208