Theorem usg2spthonot0 25696

Description: A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.)

Assertion

Ref	Expression
usg2spthonot0	|- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) <-> ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) ) )

Proof of Theorem usg2spthonot0

Step	Hyp	Ref	Expression
1		ne0i 3728	. . . . 5 |- ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> ( A ( V 2SPathOnOt E ) C ) =/= (/) )
2		2spontn0vne 25694	. . . . 5 |- ( ( A ( V 2SPathOnOt E ) C ) =/= (/) -> A =/= C )
3	1, 2	syl 17	. . . 4 |- ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> A =/= C )
4		simpl 464	. . . . . . . . . . 11 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> V USGrph E )
5	4	adantl 473	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> V USGrph E )
6		3simpb 1028	. . . . . . . . . . . 12 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ C e. V ) )
7	6	adantl 473	. . . . . . . . . . 11 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A e. V /\ C e. V ) )
8	7	adantl 473	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( A e. V /\ C e. V ) )
9		simpl 464	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> A =/= C )
10		2pthwlkonot 25692	. . . . . . . . . 10 |- ( ( V USGrph E /\ ( A e. V /\ C e. V ) /\ A =/= C ) -> ( A ( V 2SPathOnOt E ) C ) = ( A ( V 2WalksOnOt E ) C ) )
11	5, 8, 9, 10	syl3anc 1292	. . . . . . . . 9 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( A ( V 2SPathOnOt E ) C ) = ( A ( V 2WalksOnOt E ) C ) )
12	11	eleq2d 2534	. . . . . . . 8 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) <-> <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) ) )
13		usgrav 25144	. . . . . . . . . . . 12 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
14	13, 6	anim12i 576	. . . . . . . . . . 11 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) )
15	14	adantl 473	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) )
16		el2wlkonotot1 25681	. . . . . . . . . 10 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) <-> ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) )
17	15, 16	syl 17	. . . . . . . . 9 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) <-> ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) )
18		df-3an 1009	. . . . . . . . 9 |- ( ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) <-> ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) )
19	17, 18	syl6bb 269	. . . . . . . 8 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) <-> ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) )
20	12, 19	bitrd 261	. . . . . . 7 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) <-> ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) )
21		simpll 768	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> S = A )
22		simpr 468	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> T = C )
23	22	adantr 472	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> T = C )
24		simpr 468	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> A =/= C )
25	21, 23, 24	3jca 1210	. . . . . . . . . . . 12 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> ( S = A /\ T = C /\ A =/= C ) )
26	25	ex 441	. . . . . . . . . . 11 |- ( ( S = A /\ T = C ) -> ( A =/= C -> ( S = A /\ T = C /\ A =/= C ) ) )
27	26	adantr 472	. . . . . . . . . 10 |- ( ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) -> ( A =/= C -> ( S = A /\ T = C /\ A =/= C ) ) )
28	27	com12 31	. . . . . . . . 9 |- ( A =/= C -> ( ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) -> ( S = A /\ T = C /\ A =/= C ) ) )
29	28	adantr 472	. . . . . . . 8 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) -> ( S = A /\ T = C /\ A =/= C ) ) )
30	5	adantl 473	. . . . . . . . . . . 12 |- ( ( ( S = A /\ T = C ) /\ ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) -> V USGrph E )
31		simprrr 783	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) -> ( A e. V /\ B e. V /\ C e. V ) )
32		eleq1 2537	. . . . . . . . . . . . . . . 16 |- ( S = A -> ( S e. V <-> A e. V ) )
33	32	adantr 472	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> ( S e. V <-> A e. V ) )
34		eleq1 2537	. . . . . . . . . . . . . . . 16 |- ( T = C -> ( T e. V <-> C e. V ) )
35	34	adantl 473	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> ( T e. V <-> C e. V ) )
36	33, 35	3anbi13d 1367	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> ( ( S e. V /\ B e. V /\ T e. V ) <-> ( A e. V /\ B e. V /\ C e. V ) ) )
37	36	adantr 472	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) -> ( ( S e. V /\ B e. V /\ T e. V ) <-> ( A e. V /\ B e. V /\ C e. V ) ) )
38	31, 37	mpbird 240	. . . . . . . . . . . 12 |- ( ( ( S = A /\ T = C ) /\ ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) -> ( S e. V /\ B e. V /\ T e. V ) )
39		usg2wlkonot 25690	. . . . . . . . . . . 12 |- ( ( V USGrph E /\ ( S e. V /\ B e. V /\ T e. V ) ) -> ( <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) <-> ( { S , B } e. ran E /\ { B , T } e. ran E ) ) )
40	30, 38, 39	syl2anc 673	. . . . . . . . . . 11 |- ( ( ( S = A /\ T = C ) /\ ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) -> ( <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) <-> ( { S , B } e. ran E /\ { B , T } e. ran E ) ) )
41		preq1 4042	. . . . . . . . . . . . . . . 16 |- ( S = A -> { S , B } = { A , B } )
42	41	adantr 472	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> { S , B } = { A , B } )
43	42	eleq1d 2533	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> ( { S , B } e. ran E <-> { A , B } e. ran E ) )
44		preq2 4043	. . . . . . . . . . . . . . . 16 |- ( T = C -> { B , T } = { B , C } )
45	44	adantl 473	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> { B , T } = { B , C } )
46	45	eleq1d 2533	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> ( { B , T } e. ran E <-> { B , C } e. ran E ) )
47	43, 46	anbi12d 725	. . . . . . . . . . . . 13 |- ( ( S = A /\ T = C ) -> ( ( { S , B } e. ran E /\ { B , T } e. ran E ) <-> ( { A , B } e. ran E /\ { B , C } e. ran E ) ) )
48	47	biimpd 212	. . . . . . . . . . . 12 |- ( ( S = A /\ T = C ) -> ( ( { S , B } e. ran E /\ { B , T } e. ran E ) -> ( { A , B } e. ran E /\ { B , C } e. ran E ) ) )
49	48	adantr 472	. . . . . . . . . . 11 |- ( ( ( S = A /\ T = C ) /\ ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) -> ( ( { S , B } e. ran E /\ { B , T } e. ran E ) -> ( { A , B } e. ran E /\ { B , C } e. ran E ) ) )
50	40, 49	sylbid 223	. . . . . . . . . 10 |- ( ( ( S = A /\ T = C ) /\ ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) -> ( <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) -> ( { A , B } e. ran E /\ { B , C } e. ran E ) ) )
51	50	impancom 447	. . . . . . . . 9 |- ( ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) -> ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( { A , B } e. ran E /\ { B , C } e. ran E ) ) )
52	51	com12 31	. . . . . . . 8 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) -> ( { A , B } e. ran E /\ { B , C } e. ran E ) ) )
53	29, 52	jcad 542	. . . . . . 7 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) -> ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) ) )
54	20, 53	sylbid 223	. . . . . 6 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) ) )
55	54	ex 441	. . . . 5 |- ( A =/= C -> ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) ) ) )
56	55	com23 80	. . . 4 |- ( A =/= C -> ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) ) ) )
57	3, 56	mpcom 36	. . 3 |- ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) ) )
58	57	com12 31	. 2 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) ) )
59		simpll 768	. . . . . . . 8 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( S = A /\ T = C /\ A =/= C ) ) -> V USGrph E )
60		eleq1 2537	. . . . . . . . . . . . . . 15 |- ( A = S -> ( A e. V <-> S e. V ) )
61	60	eqcoms 2479	. . . . . . . . . . . . . 14 |- ( S = A -> ( A e. V <-> S e. V ) )
62	61	adantr 472	. . . . . . . . . . . . 13 |- ( ( S = A /\ T = C ) -> ( A e. V <-> S e. V ) )
63		eleq1 2537	. . . . . . . . . . . . . . 15 |- ( C = T -> ( C e. V <-> T e. V ) )
64	63	eqcoms 2479	. . . . . . . . . . . . . 14 |- ( T = C -> ( C e. V <-> T e. V ) )
65	64	adantl 473	. . . . . . . . . . . . 13 |- ( ( S = A /\ T = C ) -> ( C e. V <-> T e. V ) )
66	62, 65	3anbi13d 1367	. . . . . . . . . . . 12 |- ( ( S = A /\ T = C ) -> ( ( A e. V /\ B e. V /\ C e. V ) <-> ( S e. V /\ B e. V /\ T e. V ) ) )
67	66	biimpd 212	. . . . . . . . . . 11 |- ( ( S = A /\ T = C ) -> ( ( A e. V /\ B e. V /\ C e. V ) -> ( S e. V /\ B e. V /\ T e. V ) ) )
68	67	adantld 474	. . . . . . . . . 10 |- ( ( S = A /\ T = C ) -> ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( S e. V /\ B e. V /\ T e. V ) ) )
69	68	3adant3 1050	. . . . . . . . 9 |- ( ( S = A /\ T = C /\ A =/= C ) -> ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( S e. V /\ B e. V /\ T e. V ) ) )
70	69	impcom 437	. . . . . . . 8 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( S = A /\ T = C /\ A =/= C ) ) -> ( S e. V /\ B e. V /\ T e. V ) )
71	59, 70, 39	syl2anc 673	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( S = A /\ T = C /\ A =/= C ) ) -> ( <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) <-> ( { S , B } e. ran E /\ { B , T } e. ran E ) ) )
72	47	3adant3 1050	. . . . . . . 8 |- ( ( S = A /\ T = C /\ A =/= C ) -> ( ( { S , B } e. ran E /\ { B , T } e. ran E ) <-> ( { A , B } e. ran E /\ { B , C } e. ran E ) ) )
73	72	adantl 473	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( S = A /\ T = C /\ A =/= C ) ) -> ( ( { S , B } e. ran E /\ { B , T } e. ran E ) <-> ( { A , B } e. ran E /\ { B , C } e. ran E ) ) )
74	71, 73	bitr2d 262	. . . . . 6 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( S = A /\ T = C /\ A =/= C ) ) -> ( ( { A , B } e. ran E /\ { B , C } e. ran E ) <-> <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) )
75	74	pm5.32da 653	. . . . 5 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) <-> ( ( S = A /\ T = C /\ A =/= C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) )
76		df-3an 1009	. . . . . . . 8 |- ( ( S = A /\ T = C /\ A =/= C ) <-> ( ( S = A /\ T = C ) /\ A =/= C ) )
77		ancom 457	. . . . . . . 8 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) <-> ( A =/= C /\ ( S = A /\ T = C ) ) )
78	76, 77	bitri 257	. . . . . . 7 |- ( ( S = A /\ T = C /\ A =/= C ) <-> ( A =/= C /\ ( S = A /\ T = C ) ) )
79	78	anbi1i 709	. . . . . 6 |- ( ( ( S = A /\ T = C /\ A =/= C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) <-> ( ( A =/= C /\ ( S = A /\ T = C ) ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) )
80		anass 661	. . . . . 6 |- ( ( ( A =/= C /\ ( S = A /\ T = C ) ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) <-> ( A =/= C /\ ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) )
81	18	bicomi 207	. . . . . . 7 |- ( ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) <-> ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) )
82	81	anbi2i 708	. . . . . 6 |- ( ( A =/= C /\ ( ( S = A /\ T = C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) <-> ( A =/= C /\ ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) )
83	79, 80, 82	3bitri 279	. . . . 5 |- ( ( ( S = A /\ T = C /\ A =/= C ) /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) <-> ( A =/= C /\ ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) )
84	75, 83	syl6bb 269	. . . 4 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) <-> ( A =/= C /\ ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) ) )
85	14, 16	syl 17	. . . . . 6 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) <-> ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) )
86	85	bicomd 206	. . . . 5 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) <-> <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) ) )
87	86	anbi2d 718	. . . 4 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A =/= C /\ ( S = A /\ T = C /\ <. S , B , T >. e. ( S ( V 2WalksOnOt E ) T ) ) ) <-> ( A =/= C /\ <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) ) ) )
88	84, 87	bitrd 261	. . 3 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) <-> ( A =/= C /\ <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) ) ) )
89		simpll 768	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> V USGrph E )
90	7	adantr 472	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( A e. V /\ C e. V ) )
91		simpr 468	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> A =/= C )
92	10	eqcomd 2477	. . . . . . 7 |- ( ( V USGrph E /\ ( A e. V /\ C e. V ) /\ A =/= C ) -> ( A ( V 2WalksOnOt E ) C ) = ( A ( V 2SPathOnOt E ) C ) )
93	89, 90, 91, 92	syl3anc 1292	. . . . . 6 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( A ( V 2WalksOnOt E ) C ) = ( A ( V 2SPathOnOt E ) C ) )
94	93	eleq2d 2534	. . . . 5 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) <-> <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) ) )
95	94	biimpd 212	. . . 4 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) -> <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) ) )
96	95	expimpd 614	. . 3 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A =/= C /\ <. S , B , T >. e. ( A ( V 2WalksOnOt E ) C ) ) -> <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) ) )
97	88, 96	sylbid 223	. 2 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) ) )
98	58, 97	impbid 195	1 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) <-> ( ( S = A /\ T = C /\ A =/= C ) /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   _Vcvv 3031   (/)c0 3722   {cpr 3961   <.cotp 3967   class class class wbr 4395   ran crn 4840  (class class class)co 6308   USGrph cusg 25136   2WalksOnOt c2wlkonot 25662   2SPathOnOt c2pthonot 25664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-usgra 25139  df-wlk 25315  df-trail 25316  df-pth 25317  df-spth 25318  df-wlkon 25321  df-spthon 25324  df-2wlkonot 25665  df-2spthonot 25667

This theorem is referenced by:  usg2spthonot1  25697