Theorem tgbtwnconn1 23007

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 6107	. . . . . 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 2520	. . . . 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 6107	. . . . . 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 2520	. . . . 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 2608	. . . . . 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 2443	. . . . . . . . . . 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 22934	. . . . . . . . . . 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 23006	. . . . . . . . . 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 22942	. . . . . . . . . . . 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 22942	. . . . . . . . . . . 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 22928	. . . . . . . . . . 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 2862	. . . . . . . . 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 22925	. . . . . . . . . 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 2862	. . . . . . . 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 22925	. . . . . . . . 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 2862	. . . . . . 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 22925	. . . 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 2862	. 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 22925	. 2 |- ( ph -> E. e e. P ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) )
83	81, 82	r19.29a 2862	1 |- ( ph -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2606   E.wrex 2716   ` cfv 5418  (class class class)co 6091   Basecbs 14174   distcds 14247  TarskiGcstrkg 22889  Itvcitv 22897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-s2 12475  df-s3 12476  df-trkgc 22909  df-trkgb 22910  df-trkgcb 22911  df-trkg 22916  df-cgrg 22964

This theorem is referenced by:  tgbtwnconn2  23008  tgbtwnconnln1  23011