Theorem usgr2trlncl 26656

Description: In a simple graph, any trail of length 2 does not start and end at the same vertex. (Contributed by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)

Assertion

Ref	Expression
usgr2trlncl	|- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F (Trails ` G ) P -> ( P ` 0 ) =/= ( P ` 2 ) ) )

Proof of Theorem usgr2trlncl

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrupgr 26077	. . . . 5 |- ( G e. USGraph -> G e. UPGraph )
2		eqid 2622	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2622	. . . . . 6 |- (iEdg ` G ) = (iEdg ` G )
4	2, 3	upgrf1istrl 26600	. . . . 5 |- ( G e. UPGraph -> ( F (Trails ` G ) P <-> ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
5	1, 4	syl 17	. . . 4 |- ( G e. USGraph -> ( F (Trails ` G ) P <-> ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
6		eqidd 2623	. . . . . . . . . . . 12 |- ( ( # ` F ) = 2 -> F = F )
7		oveq2 6658	. . . . . . . . . . . . 13 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
8		fzo0to2pr 12553	. . . . . . . . . . . . 13 |- ( 0..^ 2 ) = { 0 , 1 }
9	7, 8	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
10		eqidd 2623	. . . . . . . . . . . 12 |- ( ( # ` F ) = 2 -> dom (iEdg ` G ) = dom (iEdg ` G ) )
11	6, 9, 10	f1eq123d 6131	. . . . . . . . . . 11 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) <-> F : { 0 , 1 } -1-1-> dom (iEdg ` G ) ) )
12	9	raleqdv 3144	. . . . . . . . . . . 12 |- ( ( # ` F ) = 2 -> ( A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> A. i e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
13		2wlklem 26563	. . . . . . . . . . . 12 |- ( A. i e. { 0 , 1 } ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
14	12, 13	syl6bb 276	. . . . . . . . . . 11 |- ( ( # ` F ) = 2 -> ( A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) )
15	11, 14	anbi12d 747	. . . . . . . . . 10 |- ( ( # ` F ) = 2 -> ( ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) <-> ( F : { 0 , 1 } -1-1-> dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
16	15	adantl 482	. . . . . . . . 9 |- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) <-> ( F : { 0 , 1 } -1-1-> dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
17		c0ex 10034	. . . . . . . . . . . . . 14 |- 0 e. _V
18		1ex 10035	. . . . . . . . . . . . . 14 |- 1 e. _V
19	17, 18	pm3.2i 471	. . . . . . . . . . . . 13 |- ( 0 e. _V /\ 1 e. _V )
20		0ne1 11088	. . . . . . . . . . . . 13 |- 0 =/= 1
21		eqid 2622	. . . . . . . . . . . . . 14 |- { 0 , 1 } = { 0 , 1 }
22	21	f12dfv 6529	. . . . . . . . . . . . 13 |- ( ( ( 0 e. _V /\ 1 e. _V ) /\ 0 =/= 1 ) -> ( F : { 0 , 1 } -1-1-> dom (iEdg ` G ) <-> ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) )
23	19, 20, 22	mp2an 708	. . . . . . . . . . . 12 |- ( F : { 0 , 1 } -1-1-> dom (iEdg ` G ) <-> ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) )
24		eqid 2622	. . . . . . . . . . . . . 14 |- (Edg ` G ) = (Edg ` G )
25	3, 24	usgrf1oedg 26099	. . . . . . . . . . . . 13 |- ( G e. USGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> (Edg ` G ) )
26		f1of1 6136	. . . . . . . . . . . . . 14 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> (Edg ` G ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) )
27		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F : { 0 , 1 } --> dom (iEdg ` G ) -> F : { 0 , 1 } --> dom (iEdg ` G ) )
28	17	prid1 4297	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 0 e. { 0 , 1 }
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F : { 0 , 1 } --> dom (iEdg ` G ) -> 0 e. { 0 , 1 } )
30	27, 29	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( F : { 0 , 1 } --> dom (iEdg ` G ) -> ( F ` 0 ) e. dom (iEdg ` G ) )
31	18	prid2 4298	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 1 e. { 0 , 1 }
32	31	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F : { 0 , 1 } --> dom (iEdg ` G ) -> 1 e. { 0 , 1 } )
33	27, 32	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( F : { 0 , 1 } --> dom (iEdg ` G ) -> ( F ` 1 ) e. dom (iEdg ` G ) )
34	30, 33	jca 554	. . . . . . . . . . . . . . . . . . . . . 22 |- ( F : { 0 , 1 } --> dom (iEdg ` G ) -> ( ( F ` 0 ) e. dom (iEdg ` G ) /\ ( F ` 1 ) e. dom (iEdg ` G ) ) )
35	34	anim2i 593	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) /\ F : { 0 , 1 } --> dom (iEdg ` G ) ) -> ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) /\ ( ( F ` 0 ) e. dom (iEdg ` G ) /\ ( F ` 1 ) e. dom (iEdg ` G ) ) ) )
36	35	ancoms 469	. . . . . . . . . . . . . . . . . . . 20 |- ( ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) ) -> ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) /\ ( ( F ` 0 ) e. dom (iEdg ` G ) /\ ( F ` 1 ) e. dom (iEdg ` G ) ) ) )
37		f1veqaeq 6514	. . . . . . . . . . . . . . . . . . . 20 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) /\ ( ( F ` 0 ) e. dom (iEdg ` G ) /\ ( F ` 1 ) e. dom (iEdg ` G ) ) ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = ( (iEdg ` G ) ` ( F ` 1 ) ) -> ( F ` 0 ) = ( F ` 1 ) ) )
38	36, 37	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) = ( (iEdg ` G ) ` ( F ` 1 ) ) -> ( F ` 0 ) = ( F ` 1 ) ) )
39	38	necon3d 2815	. . . . . . . . . . . . . . . . . 18 |- ( ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) ) -> ( ( F ` 0 ) =/= ( F ` 1 ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) =/= ( (iEdg ` G ) ` ( F ` 1 ) ) ) )
40		simpl 473	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } )
41		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } )
42	40, 41	neeq12d 2855	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) =/= ( (iEdg ` G ) ` ( F ` 1 ) ) <-> { ( P ` 0 ) , ( P ` 1 ) } =/= { ( P ` 1 ) , ( P ` 2 ) } ) )
43		preq1 4268	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( P ` 0 ) = ( P ` 2 ) -> { ( P ` 0 ) , ( P ` 1 ) } = { ( P ` 2 ) , ( P ` 1 ) } )
44		prcom 4267	. . . . . . . . . . . . . . . . . . . . . . 23 |- { ( P ` 2 ) , ( P ` 1 ) } = { ( P ` 1 ) , ( P ` 2 ) }
45	43, 44	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( P ` 0 ) = ( P ` 2 ) -> { ( P ` 0 ) , ( P ` 1 ) } = { ( P ` 1 ) , ( P ` 2 ) } )
46	45	necon3i 2826	. . . . . . . . . . . . . . . . . . . . 21 |- ( { ( P ` 0 ) , ( P ` 1 ) } =/= { ( P ` 1 ) , ( P ` 2 ) } -> ( P ` 0 ) =/= ( P ` 2 ) )
47	42, 46	syl6bi 243	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( (iEdg ` G ) ` ( F ` 0 ) ) =/= ( (iEdg ` G ) ` ( F ` 1 ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
48	47	com12 32	. . . . . . . . . . . . . . . . . . 19 |- ( ( (iEdg ` G ) ` ( F ` 0 ) ) =/= ( (iEdg ` G ) ` ( F ` 1 ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
49	48	a1d 25	. . . . . . . . . . . . . . . . . 18 |- ( ( (iEdg ` G ) ` ( F ` 0 ) ) =/= ( (iEdg ` G ) ` ( F ` 1 ) ) -> ( G e. USGraph -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
50	39, 49	syl6 35	. . . . . . . . . . . . . . . . 17 |- ( ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) ) -> ( ( F ` 0 ) =/= ( F ` 1 ) -> ( G e. USGraph -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) ) )
51	50	expcom 451	. . . . . . . . . . . . . . . 16 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) -> ( F : { 0 , 1 } --> dom (iEdg ` G ) -> ( ( F ` 0 ) =/= ( F ` 1 ) -> ( G e. USGraph -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) ) ) )
52	51	impd 447	. . . . . . . . . . . . . . 15 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) -> ( ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) -> ( G e. USGraph -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) ) )
53	52	com23 86	. . . . . . . . . . . . . 14 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) -> ( G e. USGraph -> ( ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) ) )
54	26, 53	syl 17	. . . . . . . . . . . . 13 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> (Edg ` G ) -> ( G e. USGraph -> ( ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) ) )
55	25, 54	mpcom 38	. . . . . . . . . . . 12 |- ( G e. USGraph -> ( ( F : { 0 , 1 } --> dom (iEdg ` G ) /\ ( F ` 0 ) =/= ( F ` 1 ) ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
56	23, 55	syl5bi 232	. . . . . . . . . . 11 |- ( G e. USGraph -> ( F : { 0 , 1 } -1-1-> dom (iEdg ` G ) -> ( ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
57	56	impd 447	. . . . . . . . . 10 |- ( G e. USGraph -> ( ( F : { 0 , 1 } -1-1-> dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
58	57	adantr 481	. . . . . . . . 9 |- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( ( F : { 0 , 1 } -1-1-> dom (iEdg ` G ) /\ ( ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( (iEdg ` G ) ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
59	16, 58	sylbid 230	. . . . . . . 8 |- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
60	59	com12 32	. . . . . . 7 |- ( ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
61	60	3adant2 1080	. . . . . 6 |- ( ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( P ` 0 ) =/= ( P ` 2 ) ) )
62	61	expdcom 455	. . . . 5 |- ( G e. USGraph -> ( ( # ` F ) = 2 -> ( ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
63	62	com23 86	. . . 4 |- ( G e. USGraph -> ( ( F : ( 0..^ ( # ` F ) ) -1-1-> dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
64	5, 63	sylbid 230	. . 3 |- ( G e. USGraph -> ( F (Trails ` G ) P -> ( ( # ` F ) = 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
65	64	com23 86	. 2 |- ( G e. USGraph -> ( ( # ` F ) = 2 -> ( F (Trails ` G ) P -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
66	65	imp 445	1 |- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F (Trails ` G ) P -> ( P ` 0 ) =/= ( P ` 2 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   {cpr 4179   class class class wbr 4653   dom cdm 5114   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   ...cfz 12326  ..^cfzo 12465   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UPGraph cupgr 25975   USGraph cusgr 26044  Trailsctrls 26587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046  df-wlks 26495  df-trls 26589

This theorem is referenced by:  usgr2trlspth  26657  usgr2trlncrct  26698