Theorem tgbtwnconn1 23827

Description: Connectitivy 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 758	. . . . . . . 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 459	. . . . . . 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 465	. . . . . 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 461	. . . . . . 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 6311	. . . . . 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 2558	. . . . 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 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 ) -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )
8		simprl 755	. . . . . . 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 465	. . . . . 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 461	. . . . . . 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 6311	. . . . . 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 2558	. . . . 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 392	. . . 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 2664	. . . . . 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 731	. . . . . . . . . . . 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 737	. . . . . . . . . . 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 731	. . . . . . . . . . . 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 737	. . . . . . . . . . 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 731	. . . . . . . . . . . 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 737	. . . . . . . . . . 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 731	. . . . . . . . . . . 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 737	. . . . . . . . . . 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 731	. . . . . . . . . . . 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 737	. . . . . . . . . . 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 779	. . . . . . . . . . . 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 16	. . . . . . . . . . 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 16	. . . . . . . . . . 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 16	. . . . . . . . . . 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 2467	. . . . . . . . . . 11 |- ( dist ` G ) = ( dist ` G )
40		simp-4r 766	. . . . . . . . . . . 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 737	. . . . . . . . . . 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 754	. . . . . . . . . . . 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 737	. . . . . . . . . . 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 770	. . . . . . . . . . 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 766	. . . . . . . . . . 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 737	. . . . . . . . . . 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 737	. . . . . . . . . . 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 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 ) /\ 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 459	. . . . . . . . . . 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 758	. . . . . . . . . . . 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 459	. . . . . . . . . . 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 463	. . . . . . . . . . . . 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 737	. . . . . . . . . . . 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 23737	. . . . . . . . . . 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 756	. . . . . . . . . . . 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 737	. . . . . . . . . . 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 463	. . . . . . . . . . 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 463	. . . . . . . . . . 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 754	. . . . . . . . . . 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 755	. . . . . . . . . . 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 756	. . . . . . . . . . 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 772	. . . . . . . . . . 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 23826	. . . . . . . . . 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 23745	. . . . . . . . . . . 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 23745	. . . . . . . . . . . 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 23730	. . . . . . . . . . 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 733	. . . . . . . . . 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 3008	. . . . . . . . 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 23727	. . . . . . . . . 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 729	. . . . . . . . 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 3008	. . . . . . . 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 23727	. . . . . . . . 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 465	. . . . . . . 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 3008	. . . . . . 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 434	. . . . . 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 218	. . . . 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 378	. . . 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 784	. . 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 23727	. . . 4 |- ( ph -> E. f e. P ( C e. ( A I f ) /\ ( C ( dist ` G ) f ) = ( C ( dist ` G ) D ) ) )
80	79	ad2antrr 725	. . 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 3008	. 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 23727	. 2 |- ( ph -> E. e e. P ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) )
83	81, 82	r19.29a 3008	1 |- ( ph -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   E.wrex 2818   ` cfv 5594  (class class class)co 6295   Basecbs 14507   distcds 14581  TarskiGcstrkg 23691  Itvcitv 23698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-s2 12793  df-s3 12794  df-trkgc 23710  df-trkgb 23711  df-trkgcb 23712  df-trkg 23716  df-cgrg 23769

This theorem is referenced by:  tgbtwnconn2  23828  tgbtwnconnln1  23832