Theorem tgbtwnconn1 25470

Description: Connectivity law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. In earlier presentations of Tarski's axioms, this theorem appeared as an additional axiom. It was derived from the other axioms by Gupta, 1965. (Contributed by Thierry Arnoux, 30-Apr-2019.)

Hypotheses

Ref	Expression
tgbtwnconn1.p	|- P = ( Base ` G )
tgbtwnconn1.i	|- I = (Itv ` G )
tgbtwnconn1.g	|- ( ph -> G e. TarskiG )
tgbtwnconn1.a	|- ( ph -> A e. P )
tgbtwnconn1.b	|- ( ph -> B e. P )
tgbtwnconn1.c	|- ( ph -> C e. P )
tgbtwnconn1.d	|- ( ph -> D e. P )
tgbtwnconn1.1	|- ( ph -> A =/= B )
tgbtwnconn1.2	|- ( ph -> B e. ( A I C ) )
tgbtwnconn1.3	|- ( ph -> B e. ( A I D ) )

Assertion

Ref	Expression
tgbtwnconn1	|- ( ph -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )

Proof of Theorem tgbtwnconn1

Dummy variables e f h j x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpllr 799	. . . . . . . 8 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) )
2	1	simpld 475	. . . . . . 7 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> D e. ( A I e ) )
3	2	adantr 481	. . . . . 6 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C = e ) -> D e. ( A I e ) )
4		simpr 477	. . . . . . 7 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C = e ) -> C = e )
5	4	oveq2d 6666	. . . . . 6 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C = e ) -> ( A I C ) = ( A I e ) )
6	3, 5	eleqtrrd 2704	. . . . 5 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C = e ) -> D e. ( A I C ) )
7	6	olcd 408	. . . 4 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C = e ) -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )
8		simprl 794	. . . . . . 7 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> C e. ( A I f ) )
9	8	adantr 481	. . . . . 6 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ D = f ) -> C e. ( A I f ) )
10		simpr 477	. . . . . . 7 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ D = f ) -> D = f )
11	10	oveq2d 6666	. . . . . 6 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ D = f ) -> ( A I D ) = ( A I f ) )
12	9, 11	eleqtrrd 2704	. . . . 5 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ D = f ) -> C e. ( A I D ) )
13	12	orcd 407	. . . 4 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ D = f ) -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )
14		df-ne 2795	. . . . . 6 |- ( C =/= e <-> -. C = e )
15		tgbtwnconn1.p	. . . . . . . . . . 11 |- P = ( Base ` G )
16		tgbtwnconn1.i	. . . . . . . . . . 11 |- I = (Itv ` G )
17		tgbtwnconn1.g	. . . . . . . . . . . . 13 |- ( ph -> G e. TarskiG )
18	17	ad4antr 768	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> G e. TarskiG )
19	18	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> G e. TarskiG )
20		tgbtwnconn1.a	. . . . . . . . . . . . 13 |- ( ph -> A e. P )
21	20	ad4antr 768	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> A e. P )
22	21	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> A e. P )
23		tgbtwnconn1.b	. . . . . . . . . . . . 13 |- ( ph -> B e. P )
24	23	ad4antr 768	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> B e. P )
25	24	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> B e. P )
26		tgbtwnconn1.c	. . . . . . . . . . . . 13 |- ( ph -> C e. P )
27	26	ad4antr 768	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> C e. P )
28	27	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> C e. P )
29		tgbtwnconn1.d	. . . . . . . . . . . . 13 |- ( ph -> D e. P )
30	29	ad4antr 768	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> D e. P )
31	30	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> D e. P )
32		simp-11l 820	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> ph )
33		tgbtwnconn1.1	. . . . . . . . . . . 12 |- ( ph -> A =/= B )
34	32, 33	syl 17	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> A =/= B )
35		tgbtwnconn1.2	. . . . . . . . . . . 12 |- ( ph -> B e. ( A I C ) )
36	32, 35	syl 17	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> B e. ( A I C ) )
37		tgbtwnconn1.3	. . . . . . . . . . . 12 |- ( ph -> B e. ( A I D ) )
38	32, 37	syl 17	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> B e. ( A I D ) )
39		eqid 2622	. . . . . . . . . . 11 |- ( dist ` G ) = ( dist ` G )
40		simp-4r 807	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> e e. P )
41	40	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> e e. P )
42		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> f e. P )
43	42	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> f e. P )
44		simp-6r 811	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> h e. P )
45		simp-4r 807	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> j e. P )
46	2	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> D e. ( A I e ) )
47	8	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> C e. ( A I f ) )
48		simp-5r 809	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) )
49	48	simpld 475	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> e e. ( A I h ) )
50		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) )
51	50	simpld 475	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> f e. ( A I j ) )
52	1	simprd 479	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) )
53	52	ad7antr 774	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) )
54	15, 39, 16, 19, 31, 41, 31, 28, 53	tgcgrcomlr 25375	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> ( e ( dist ` G ) D ) = ( C ( dist ` G ) D ) )
55		simprr 796	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) )
56	55	ad7antr 774	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) )
57	48	simprd 479	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) )
58	50	simprd 479	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) )
59		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> x e. P )
60		simprl 794	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> x e. ( C I e ) )
61		simprr 796	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> x e. ( D I f ) )
62		simp-7r 813	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> C =/= e )
63	15, 16, 19, 22, 25, 28, 31, 34, 36, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 54, 56, 57, 58, 59, 60, 61, 62	tgbtwnconn1lem3 25469	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) /\ x e. P ) /\ ( x e. ( C I e ) /\ x e. ( D I f ) ) ) -> D = f )
64	15, 39, 16, 18, 21, 27, 42, 8	tgbtwncom 25383	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> C e. ( f I A ) )
65	15, 39, 16, 18, 21, 30, 40, 2	tgbtwncom 25383	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> D e. ( e I A ) )
66	15, 39, 16, 18, 42, 40, 21, 27, 30, 64, 65	axtgpasch 25366	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> E. x e. P ( x e. ( C I e ) /\ x e. ( D I f ) ) )
67	66	ad5antr 770	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) -> E. x e. P ( x e. ( C I e ) /\ x e. ( D I f ) ) )
68	63, 67	r19.29a 3078	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) /\ j e. P ) /\ ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) ) -> D = f )
69	15, 39, 16, 18, 21, 42, 24, 30	axtgsegcon 25363	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> E. j e. P ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) )
70	69	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) -> E. j e. P ( f e. ( A I j ) /\ ( f ( dist ` G ) j ) = ( B ( dist ` G ) D ) ) )
71	68, 70	r19.29a 3078	. . . . . . . 8 |- ( ( ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) /\ h e. P ) /\ ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) ) -> D = f )
72	15, 39, 16, 18, 21, 40, 24, 27	axtgsegcon 25363	. . . . . . . . 9 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> E. h e. P ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) )
73	72	adantr 481	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) -> E. h e. P ( e e. ( A I h ) /\ ( e ( dist ` G ) h ) = ( B ( dist ` G ) C ) ) )
74	71, 73	r19.29a 3078	. . . . . . 7 |- ( ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) /\ C =/= e ) -> D = f )
75	74	ex 450	. . . . . 6 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> ( C =/= e -> D = f ) )
76	14, 75	syl5bir 233	. . . . 5 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> ( -. C = e -> D = f ) )
77	76	orrd 393	. . . 4 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> ( C = e \/ D = f ) )
78	7, 13, 77	mpjaodan 827	. . 3 |- ( ( ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) /\ f e. P ) /\ ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) ) -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )
79	15, 39, 16, 17, 20, 26, 26, 29	axtgsegcon 25363	. . . 4 |- ( ph -> E. f e. P ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) )
80	79	ad2antrr 762	. . 3 |- ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) -> E. f e. P ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) )
81	78, 80	r19.29a 3078	. 2 |- ( ( ( ph /\ e e. P ) /\ ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) ) -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )
82	15, 39, 16, 17, 20, 29, 29, 26	axtgsegcon 25363	. 2 |- ( ph -> E. e e. P ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) )
83	81, 82	r19.29a 3078	1 |- ( ph -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406

This theorem is referenced by:  tgbtwnconn2  25471  tgbtwnconnln1  25475  hltr  25505  hlbtwn  25506