Theorem usg2spthonot0 24754

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 3773	. . . . 5 |- ( <. S , B , T >. e. ( A ( V 2SPathOnOt E ) C ) -> ( A ( V 2SPathOnOt E ) C ) =/= (/) )
2		2spontn0vne 24752	. . . . 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 993	. . . . . . . . . . . 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 24750	. . . . . . . . . 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 1227	. . . . . . . . 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 2511	. . . . . . . 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 24203	. . . . . . . . . . . 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 24739	. . . . . . . . . 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 974	. . . . . . . . 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 1175	. . . . . . . . . . . 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 2513	. . . . . . . . . . . . . . . 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 2513	. . . . . . . . . . . . . . . 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 1300	. . . . . . . . . . . . . 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 24748	. . . . . . . . . . . 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 4090	. . . . . . . . . . . . . . . 16 |- ( S = A -> { S , B } = { A , B } )
42	41	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> { S , B } = { A , B } )
43	42	eleq1d 2510	. . . . . . . . . . . . . 14 |- ( ( S = A /\ T = C ) -> ( { S , B } e. ran E <-> { A , B } e. ran E ) )
44		preq2 4091	. . . . . . . . . . . . . . . 16 |- ( T = C -> { B , T } = { B , C } )
45	44	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( S = A /\ T = C ) -> { B , T } = { B , C } )
46	45	eleq1d 2510	. . . . . . . . . . . . . 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 2513	. . . . . . . . . . . . . . 15 |- ( A = S -> ( A e. V <-> S e. V ) )
61	60	eqcoms 2453	. . . . . . . . . . . . . 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 2513	. . . . . . . . . . . . . . 15 |- ( C = T -> ( C e. V <-> T e. V ) )
64	63	eqcoms 2453	. . . . . . . . . . . . . 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 1300	. . . . . . . . . . . 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 1015	. . . . . . . . 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 1015	. . . . . . . 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 974	. . . . . . . 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 2449	. . . . . . 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 1227	. . . . . 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 2511	. . . . 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 972   = wceq 1381   e. wcel 1802   =/= wne 2636   _Vcvv 3093   (/)c0 3767   {cpr 4012   <.cotp 4018   class class class wbr 4433   ran crn 4986  (class class class)co 6277   USGrph cusg 24195   2WalksOnOt c2wlkonot 24720   2SPathOnOt c2pthonot 24722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-ot 4019  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-hash 12380  df-word 12516  df-usgra 24198  df-wlk 24373  df-trail 24374  df-pth 24375  df-spth 24376  df-wlkon 24379  df-spthon 24382  df-2wlkonot 24723  df-2spthonot 24725

This theorem is referenced by:  usg2spthonot1  24755