Theorem tgbtwnconn1 24088

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 760	. . . . . . . 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 6312	. . . . . 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 2548	. . . . 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 756	. . . . . . 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 6312	. . . . . 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 2548	. . . . 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 2654	. . . . . 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 781	. . . . . . . . . . . 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 2457	. . . . . . . . . . 11 |- ( dist ` G ) = ( dist ` G )
40		simp-4r 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 ) ) ) -> 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 755	. . . . . . . . . . . 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 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 ) ) ) -> h e. P )
45		simp-4r 768	. . . . . . . . . . 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 770	. . . . . . . . . . . 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 760	. . . . . . . . . . . 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 23997	. . . . . . . . . . 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 757	. . . . . . . . . . . 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 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. P )
60		simprl 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. ( C I e ) )
61		simprr 757	. . . . . . . . . . 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 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 )
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 24087	. . . . . . . . . 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 24005	. . . . . . . . . . . 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 24005	. . . . . . . . . . . 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 23990	. . . . . . . . . . 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 2999	. . . . . . . . 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 23987	. . . . . . . . . 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 2999	. . . . . . . 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 23987	. . . . . . . . 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 2999	. . . . . . 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 786	. . 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 23987	. . . 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 2999	. 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 23987	. 2 |- ( ph -> E. e e. P ( D e. ( A I e ) /\ ( D ( dist ` G ) e ) = ( D ( dist ` G ) C ) ) )
83	81, 82	r19.29a 2999	1 |- ( ph -> ( C e. ( A I D ) \/ D e. ( A I C ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   E.wrex 2808   ` cfv 5594  (class class class)co 6296   Basecbs 14644   distcds 14721  TarskiGcstrkg 23951  Itvcitv 23958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976  df-cgrg 24029

This theorem is referenced by:  tgbtwnconn2  24089  tgbtwnconnln1  24093  hltr  24120  hlbtwn  24121