Theorem usg2spthonot0 25617

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 3737	. . . . 5 |- ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> ( A ( V 2SPathOnOt E ) C ) =/= (/) )
2		2spontn0vne 25615	. . . . 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 459	. . . . . . . . . . 11 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> V USGrph E )
5	4	adantl 468	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> V USGrph E )
6		3simpb 1006	. . . . . . . . . . . 12 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ C e. V ) )
7	6	adantl 468	. . . . . . . . . . 11 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A e. V /\ C e. V ) )
8	7	adantl 468	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( A e. V /\ C e. V ) )
9		simpl 459	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> A =/= C )
10		2pthwlkonot 25613	. . . . . . . . . 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 1268	. . . . . . . . 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 2514	. . . . . . . 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 25065	. . . . . . . . . . . 12 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
14	13, 6	anim12i 570	. . . . . . . . . . 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 468	. . . . . . . . . 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 25602	. . . . . . . . . 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 987	. . . . . . . . 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 265	. . . . . . . 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 257	. . . . . . 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 760	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> S = A )
22		simpr 463	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> T = C )
23	22	adantr 467	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> T = C )
24		simpr 463	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> A =/= C )
25	21, 23, 24	3jca 1188	. . . . . . . . . . . 12 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> ( S = A /\ T = C /\ A =/= C ) )
26	25	ex 436	. . . . . . . . . . 11 |- ( ( S = A /\ T = C ) -> ( A =/= C -> ( S = A /\ T = C /\ A =/= C ) ) )
27	26	adantr 467	. . . . . . . . . 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 32	. . . . . . . . 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 467	. . . . . . . 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 468	. . . . . . . . . . . 12 |- ( ( ( S = A /\ T = C ) /\ ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) -> V USGrph E )
31		simprrr 775	. . . . . . . . . . . . 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 2517	. . . . . . . . . . . . . . . 16 |- ( S = A -> ( S e. V <-> A e. V ) )
33	32	adantr 467	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> ( S e. V <-> A e. V ) )
34		eleq1 2517	. . . . . . . . . . . . . . . 16 |- ( T = C -> ( T e. V <-> C e. V ) )
35	34	adantl 468	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> ( T e. V <-> C e. V ) )
36	33, 35	3anbi13d 1341	. . . . . . . . . . . . . 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 467	. . . . . . . . . . . . 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 236	. . . . . . . . . . . 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 25611	. . . . . . . . . . . 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 667	. . . . . . . . . . 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 4051	. . . . . . . . . . . . . . . 16 |- ( S = A -> { S , B } = { A , B } )
42	41	adantr 467	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> { S , B } = { A , B } )
43	42	eleq1d 2513	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> ( { S , B } e. ran E <-> { A , B } e. ran E ) )
44		preq2 4052	. . . . . . . . . . . . . . . 16 |- ( T = C -> { B , T } = { B , C } )
45	44	adantl 468	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> { B , T } = { B , C } )
46	45	eleq1d 2513	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> ( { B , T } e. ran E <-> { B , C } e. ran E ) )
47	43, 46	anbi12d 717	. . . . . . . . . . . . 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 211	. . . . . . . . . . . 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 467	. . . . . . . . . . 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 219	. . . . . . . . . 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 442	. . . . . . . . 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 32	. . . . . . . 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 536	. . . . . . 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 219	. . . . . 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 436	. . . . 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 81	. . . 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 37	. . 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 32	. 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 760	. . . . . . . 8 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( S = A /\ T = C /\ A =/= C ) ) -> V USGrph E )
60		eleq1 2517	. . . . . . . . . . . . . . 15 |- ( A = S -> ( A e. V <-> S e. V ) )
61	60	eqcoms 2459	. . . . . . . . . . . . . 14 |- ( S = A -> ( A e. V <-> S e. V ) )
62	61	adantr 467	. . . . . . . . . . . . 13 |- ( ( S = A /\ T = C ) -> ( A e. V <-> S e. V ) )
63		eleq1 2517	. . . . . . . . . . . . . . 15 |- ( C = T -> ( C e. V <-> T e. V ) )
64	63	eqcoms 2459	. . . . . . . . . . . . . 14 |- ( T = C -> ( C e. V <-> T e. V ) )
65	64	adantl 468	. . . . . . . . . . . . 13 |- ( ( S = A /\ T = C ) -> ( C e. V <-> T e. V ) )
66	62, 65	3anbi13d 1341	. . . . . . . . . . . 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 211	. . . . . . . . . . 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 469	. . . . . . . . . 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 1028	. . . . . . . . 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 432	. . . . . . . 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 667	. . . . . . 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 1028	. . . . . . . 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 468	. . . . . . 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 258	. . . . . 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 647	. . . . 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 987	. . . . . . . 8 |- ( ( S = A /\ T = C /\ A =/= C ) <-> ( ( S = A /\ T = C ) /\ A =/= C ) )
77		ancom 452	. . . . . . . 8 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) <-> ( A =/= C /\ ( S = A /\ T = C ) ) )
78	76, 77	bitri 253	. . . . . . 7 |- ( ( S = A /\ T = C /\ A =/= C ) <-> ( A =/= C /\ ( S = A /\ T = C ) ) )
79	78	anbi1i 701	. . . . . 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 655	. . . . . 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 206	. . . . . . 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 700	. . . . . 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 275	. . . . 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 265	. . . 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 205	. . . . 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 710	. . . 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 257	. . 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 760	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> V USGrph E )
90	7	adantr 467	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( A e. V /\ C e. V ) )
91		simpr 463	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> A =/= C )
92	10	eqcomd 2457	. . . . . . 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 1268	. . . . . 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 2514	. . . . 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 211	. . . 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 608	. . 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 219	. 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 194	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   _Vcvv 3045   (/)c0 3731   {cpr 3970   <.cotp 3976   class class class wbr 4402   ran crn 4835  (class class class)co 6290   USGrph cusg 25057   2WalksOnOt c2wlkonot 25583   2SPathOnOt c2pthonot 25585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-usgra 25060  df-wlk 25236  df-trail 25237  df-pth 25238  df-spth 25239  df-wlkon 25242  df-spthon 25245  df-2wlkonot 25586  df-2spthonot 25588

This theorem is referenced by:  usg2spthonot1  25618