Theorem usg2spthonot0 24553

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 3786	. . . . 5 |- ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> ( A ( V 2SPathOnOt E ) C ) =/= (/) )
2		2spontn0vne 24551	. . . . 5 |- ( ( A ( V 2SPathOnOt E ) C ) =/= (/) -> A =/= C )
3	1, 2	syl 16	. . . 4 |- ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> A =/= C )
4		simpl 457	. . . . . . . . . . 11 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> V USGrph E )
5	4	adantl 466	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> V USGrph E )
6		3simpb 989	. . . . . . . . . . . 12 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ C e. V ) )
7	6	adantl 466	. . . . . . . . . . 11 |- ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A e. V /\ C e. V ) )
8	7	adantl 466	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> ( A e. V /\ C e. V ) )
9		simpl 457	. . . . . . . . . 10 |- ( ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) -> A =/= C )
10		2pthwlkonot 24549	. . . . . . . . . 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 1223	. . . . . . . . 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 2532	. . . . . . . 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 24003	. . . . . . . . . . . 12 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
14	13, 6	anim12i 566	. . . . . . . . . . 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 466	. . . . . . . . . 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 24538	. . . . . . . . . 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 16	. . . . . . . . 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 970	. . . . . . . . 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 261	. . . . . . . 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 253	. . . . . . 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 753	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> S = A )
22		simpr 461	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> T = C )
23	22	adantr 465	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> T = C )
24		simpr 461	. . . . . . . . . . . . 13 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> A =/= C )
25	21, 23, 24	3jca 1171	. . . . . . . . . . . 12 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) -> ( S = A /\ T = C /\ A =/= C ) )
26	25	ex 434	. . . . . . . . . . 11 |- ( ( S = A /\ T = C ) -> ( A =/= C -> ( S = A /\ T = C /\ A =/= C ) ) )
27	26	adantr 465	. . . . . . . . . 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 465	. . . . . . . 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 466	. . . . . . . . . . . 12 |- ( ( ( S = A /\ T = C ) /\ ( A =/= C /\ ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) -> V USGrph E )
31		simprrr 764	. . . . . . . . . . . . 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 2534	. . . . . . . . . . . . . . . 16 |- ( S = A -> ( S e. V <-> A e. V ) )
33	32	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> ( S e. V <-> A e. V ) )
34		eleq1 2534	. . . . . . . . . . . . . . . 16 |- ( T = C -> ( T e. V <-> C e. V ) )
35	34	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> ( T e. V <-> C e. V ) )
36	33, 35	3anbi13d 1296	. . . . . . . . . . . . . 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 465	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 24547	. . . . . . . . . . . 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 661	. . . . . . . . . . 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 4101	. . . . . . . . . . . . . . . 16 |- ( S = A -> { S , B } = { A , B } )
42	41	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> { S , B } = { A , B } )
43	42	eleq1d 2531	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> ( { S , B } e. ran E <-> { A , B } e. ran E ) )
44		preq2 4102	. . . . . . . . . . . . . . . 16 |- ( T = C -> { B , T } = { B , C } )
45	44	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> { B , T } = { B , C } )
46	45	eleq1d 2531	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> ( { B , T } e. ran E <-> { B , C } e. ran E ) )
47	43, 46	anbi12d 710	. . . . . . . . . . . . 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 207	. . . . . . . . . . . 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 465	. . . . . . . . . . 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 215	. . . . . . . . . 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 440	. . . . . . . . 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 533	. . . . . . 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 215	. . . . . 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 434	. . . . 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 78	. . . 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 753	. . . . . . . 8 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( S = A /\ T = C /\ A =/= C ) ) -> V USGrph E )
60		eleq1 2534	. . . . . . . . . . . . . . 15 |- ( A = S -> ( A e. V <-> S e. V ) )
61	60	eqcoms 2474	. . . . . . . . . . . . . 14 |- ( S = A -> ( A e. V <-> S e. V ) )
62	61	adantr 465	. . . . . . . . . . . . 13 |- ( ( S = A /\ T = C ) -> ( A e. V <-> S e. V ) )
63		eleq1 2534	. . . . . . . . . . . . . . 15 |- ( C = T -> ( C e. V <-> T e. V ) )
64	63	eqcoms 2474	. . . . . . . . . . . . . 14 |- ( T = C -> ( C e. V <-> T e. V ) )
65	64	adantl 466	. . . . . . . . . . . . 13 |- ( ( S = A /\ T = C ) -> ( C e. V <-> T e. V ) )
66	62, 65	3anbi13d 1296	. . . . . . . . . . . 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 207	. . . . . . . . . . 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 467	. . . . . . . . . 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 1011	. . . . . . . . 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 430	. . . . . . . 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 661	. . . . . . 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 1011	. . . . . . . 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 466	. . . . . . 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 254	. . . . . 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 641	. . . . 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 970	. . . . . . . 8 |- ( ( S = A /\ T = C /\ A =/= C ) <-> ( ( S = A /\ T = C ) /\ A =/= C ) )
77		ancom 450	. . . . . . . 8 |- ( ( ( S = A /\ T = C ) /\ A =/= C ) <-> ( A =/= C /\ ( S = A /\ T = C ) ) )
78	76, 77	bitri 249	. . . . . . 7 |- ( ( S = A /\ T = C /\ A =/= C ) <-> ( A =/= C /\ ( S = A /\ T = C ) ) )
79	78	anbi1i 695	. . . . . 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 649	. . . . . 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 202	. . . . . . 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 694	. . . . . 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 271	. . . . 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 261	. . . 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 16	. . . . . 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 201	. . . . 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 703	. . . 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 253	. . 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 753	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> V USGrph E )
90	7	adantr 465	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( A e. V /\ C e. V ) )
91		simpr 461	. . . . . . 7 |- ( ( ( V USGrph E /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> A =/= C )
92	10	eqcomd 2470	. . . . . . 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 1223	. . . . . 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 2532	. . . . 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 207	. . . 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 603	. . 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 215	. 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 191	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 184   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   =/= wne 2657   _Vcvv 3108   (/)c0 3780   {cpr 4024   <.cotp 4030   class class class wbr 4442   ran crn 4995  (class class class)co 6277   USGrph cusg 23995   2WalksOnOt c2wlkonot 24519   2SPathOnOt c2pthonot 24521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-ot 4031  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-usgra 23998  df-wlk 24172  df-trail 24173  df-pth 24174  df-spth 24175  df-wlkon 24178  df-spthon 24181  df-2wlkonot 24522  df-2spthonot 24524

This theorem is referenced by:  usg2spthonot1  24554