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Theorem frgr3vlem1 27137
Description: Lemma 1 for frgr3v 27139. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
Hypotheses
Ref Expression
frgr3v.v |- V = (Vtx ` G )
frgr3v.e |- E = (Edg ` G )
Assertion
Ref Expression
frgr3vlem1 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A. x A. y ( ( ( x e. { A , B , C } /\ { { x , A } , { x , B } } C_ E ) /\ ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ E ) ) -> x = y ) )
Distinct variable groups:   x, A, y   x, B, y   x, C, y   x, E, y   x, G, y   x, V, y   x, X, y   x, Y, y   x, Z, y
Proof of Theorem frgr3vlem1
StepHypRef Expression
1 vex 3203 . . . . . 6 |- x e. _V
21eltp 4230 . . . . 5 |- ( x e. { A , B , C } <-> ( x = A \/ x = B \/ x = C ) )
3 vex 3203 . . . . . . . . 9 |- y e. _V
43eltp 4230 . . . . . . . 8 |- ( y e. { A , B , C } <-> ( y = A \/ y = B \/ y = C ) )
5 eqidd 2623 . . . . . . . . . . . . . . 15 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = A )
65a1i 11 . . . . . . . . . . . . . 14 |- ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = A ) )
76a1i13 27 . . . . . . . . . . . . 13 |- ( y = A -> ( { { A , A } , { A , B } } C_ E -> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = A ) ) ) )
8 preq1 4268 . . . . . . . . . . . . . . 15 |- ( y = A -> { y , A } = { A , A } )
9 preq1 4268 . . . . . . . . . . . . . . 15 |- ( y = A -> { y , B } = { A , B } )
108, 9preq12d 4276 . . . . . . . . . . . . . 14 |- ( y = A -> { { y , A } , { y , B } } = { { A , A } , { A , B } } )
1110sseq1d 3632 . . . . . . . . . . . . 13 |- ( y = A -> ( { { y , A } , { y , B } } C_ E <-> { { A , A } , { A , B } } C_ E ) )
12 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = A -> ( A = y <-> A = A ) )
1312imbi2d 330 . . . . . . . . . . . . . 14 |- ( y = A -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = A ) ) )
1413imbi2d 330 . . . . . . . . . . . . 13 |- ( y = A -> ( ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) <-> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = A ) ) ) )
157, 11, 143imtr4d 283 . . . . . . . . . . . 12 |- ( y = A -> ( { { y , A } , { y , B } } C_ E -> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) ) )
16 prex 4909 . . . . . . . . . . . . . . . . . . 19 |- { A , A } e. _V
17 prex 4909 . . . . . . . . . . . . . . . . . . 19 |- { A , B } e. _V
1816, 17prss 4351 . . . . . . . . . . . . . . . . . 18 |- ( ( { A , A } e. E /\ { A , B } e. E ) <-> { { A , A } , { A , B } } C_ E )
19 frgr3v.e . . . . . . . . . . . . . . . . . . . . . . 23 |- E = (Edg ` G )
2019usgredgne 26098 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( G e. USGraph /\ { A , A } e. E ) -> A =/= A )
2120adantll 750 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( V = { A , B , C } /\ G e. USGraph ) /\ { A , A } e. E ) -> A =/= A )
22 df-ne 2795 . . . . . . . . . . . . . . . . . . . . . 22 |- ( A =/= A <-> -. A = A )
23 eqid 2622 . . . . . . . . . . . . . . . . . . . . . . 23 |- A = A
2423pm2.24i 146 . . . . . . . . . . . . . . . . . . . . . 22 |- ( -. A = A -> A = B )
2522, 24sylbi 207 . . . . . . . . . . . . . . . . . . . . 21 |- ( A =/= A -> A = B )
2621, 25syl 17 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( V = { A , B , C } /\ G e. USGraph ) /\ { A , A } e. E ) -> A = B )
2726expcom 451 . . . . . . . . . . . . . . . . . . 19 |- ( { A , A } e. E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> A = B ) )
2827adantr 481 . . . . . . . . . . . . . . . . . 18 |- ( ( { A , A } e. E /\ { A , B } e. E ) -> ( ( V = { A , B , C } /\ G e. USGraph ) -> A = B ) )
2918, 28sylbir 225 . . . . . . . . . . . . . . . . 17 |- ( { { A , A } , { A , B } } C_ E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> A = B ) )
3029com12 32 . . . . . . . . . . . . . . . 16 |- ( ( V = { A , B , C } /\ G e. USGraph ) -> ( { { A , A } , { A , B } } C_ E -> A = B ) )
31303ad2ant3 1084 . . . . . . . . . . . . . . 15 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> ( { { A , A } , { A , B } } C_ E -> A = B ) )
3231com12 32 . . . . . . . . . . . . . 14 |- ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = B ) )
33322a1i 12 . . . . . . . . . . . . 13 |- ( y = B -> ( { { B , A } , { B , B } } C_ E -> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = B ) ) ) )
34 preq1 4268 . . . . . . . . . . . . . . 15 |- ( y = B -> { y , A } = { B , A } )
35 preq1 4268 . . . . . . . . . . . . . . 15 |- ( y = B -> { y , B } = { B , B } )
3634, 35preq12d 4276 . . . . . . . . . . . . . 14 |- ( y = B -> { { y , A } , { y , B } } = { { B , A } , { B , B } } )
3736sseq1d 3632 . . . . . . . . . . . . 13 |- ( y = B -> ( { { y , A } , { y , B } } C_ E <-> { { B , A } , { B , B } } C_ E ) )
38 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = B -> ( A = y <-> A = B ) )
3938imbi2d 330 . . . . . . . . . . . . . 14 |- ( y = B -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = B ) ) )
4039imbi2d 330 . . . . . . . . . . . . 13 |- ( y = B -> ( ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) <-> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = B ) ) ) )
4133, 37, 403imtr4d 283 . . . . . . . . . . . 12 |- ( y = B -> ( { { y , A } , { y , B } } C_ E -> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) ) )
4223pm2.24i 146 . . . . . . . . . . . . . . . . . . . . . 22 |- ( -. A = A -> A = C )
4322, 42sylbi 207 . . . . . . . . . . . . . . . . . . . . 21 |- ( A =/= A -> A = C )
4421, 43syl 17 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( V = { A , B , C } /\ G e. USGraph ) /\ { A , A } e. E ) -> A = C )
4544expcom 451 . . . . . . . . . . . . . . . . . . 19 |- ( { A , A } e. E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> A = C ) )
4645adantr 481 . . . . . . . . . . . . . . . . . 18 |- ( ( { A , A } e. E /\ { A , B } e. E ) -> ( ( V = { A , B , C } /\ G e. USGraph ) -> A = C ) )
4718, 46sylbir 225 . . . . . . . . . . . . . . . . 17 |- ( { { A , A } , { A , B } } C_ E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> A = C ) )
4847com12 32 . . . . . . . . . . . . . . . 16 |- ( ( V = { A , B , C } /\ G e. USGraph ) -> ( { { A , A } , { A , B } } C_ E -> A = C ) )
49483ad2ant3 1084 . . . . . . . . . . . . . . 15 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> ( { { A , A } , { A , B } } C_ E -> A = C ) )
5049com12 32 . . . . . . . . . . . . . 14 |- ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = C ) )
51502a1i 12 . . . . . . . . . . . . 13 |- ( y = C -> ( { { C , A } , { C , B } } C_ E -> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = C ) ) ) )
52 preq1 4268 . . . . . . . . . . . . . . 15 |- ( y = C -> { y , A } = { C , A } )
53 preq1 4268 . . . . . . . . . . . . . . 15 |- ( y = C -> { y , B } = { C , B } )
5452, 53preq12d 4276 . . . . . . . . . . . . . 14 |- ( y = C -> { { y , A } , { y , B } } = { { C , A } , { C , B } } )
5554sseq1d 3632 . . . . . . . . . . . . 13 |- ( y = C -> ( { { y , A } , { y , B } } C_ E <-> { { C , A } , { C , B } } C_ E ) )
56 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = C -> ( A = y <-> A = C ) )
5756imbi2d 330 . . . . . . . . . . . . . 14 |- ( y = C -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = C ) ) )
5857imbi2d 330 . . . . . . . . . . . . 13 |- ( y = C -> ( ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) <-> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = C ) ) ) )
5951, 55, 583imtr4d 283 . . . . . . . . . . . 12 |- ( y = C -> ( { { y , A } , { y , B } } C_ E -> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) ) )
6015, 41, 593jaoi 1391 . . . . . . . . . . 11 |- ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ E -> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) ) )
61 preq1 4268 . . . . . . . . . . . . . . 15 |- ( x = A -> { x , A } = { A , A } )
62 preq1 4268 . . . . . . . . . . . . . . 15 |- ( x = A -> { x , B } = { A , B } )
6361, 62preq12d 4276 . . . . . . . . . . . . . 14 |- ( x = A -> { { x , A } , { x , B } } = { { A , A } , { A , B } } )
6463sseq1d 3632 . . . . . . . . . . . . 13 |- ( x = A -> ( { { x , A } , { x , B } } C_ E <-> { { A , A } , { A , B } } C_ E ) )
65 eqeq1 2626 . . . . . . . . . . . . . 14 |- ( x = A -> ( x = y <-> A = y ) )
6665imbi2d 330 . . . . . . . . . . . . 13 |- ( x = A -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) )
6764, 66imbi12d 334 . . . . . . . . . . . 12 |- ( x = A -> ( ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) <-> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) ) )
6867imbi2d 330 . . . . . . . . . . 11 |- ( x = A -> ( ( { { y , A } , { y , B } } C_ E -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) <-> ( { { y , A } , { y , B } } C_ E -> ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A = y ) ) ) ) )
6960, 68syl5ibr 236 . . . . . . . . . 10 |- ( x = A -> ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ E -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) ) )
70 prex 4909 . . . . . . . . . . . . . . . . . . 19 |- { B , A } e. _V
71 prex 4909 . . . . . . . . . . . . . . . . . . 19 |- { B , B } e. _V
7270, 71prss 4351 . . . . . . . . . . . . . . . . . 18 |- ( ( { B , A } e. E /\ { B , B } e. E ) <-> { { B , A } , { B , B } } C_ E )
7319usgredgne 26098 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( G e. USGraph /\ { B , B } e. E ) -> B =/= B )
7473adantll 750 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( V = { A , B , C } /\ G e. USGraph ) /\ { B , B } e. E ) -> B =/= B )
75 df-ne 2795 . . . . . . . . . . . . . . . . . . . . . 22 |- ( B =/= B <-> -. B = B )
76 eqid 2622 . . . . . . . . . . . . . . . . . . . . . . 23 |- B = B
7776pm2.24i 146 . . . . . . . . . . . . . . . . . . . . . 22 |- ( -. B = B -> B = A )
7875, 77sylbi 207 . . . . . . . . . . . . . . . . . . . . 21 |- ( B =/= B -> B = A )
7974, 78syl 17 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( V = { A , B , C } /\ G e. USGraph ) /\ { B , B } e. E ) -> B = A )
8079expcom 451 . . . . . . . . . . . . . . . . . . 19 |- ( { B , B } e. E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> B = A ) )
8180adantl 482 . . . . . . . . . . . . . . . . . 18 |- ( ( { B , A } e. E /\ { B , B } e. E ) -> ( ( V = { A , B , C } /\ G e. USGraph ) -> B = A ) )
8272, 81sylbir 225 . . . . . . . . . . . . . . . . 17 |- ( { { B , A } , { B , B } } C_ E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> B = A ) )
8382com12 32 . . . . . . . . . . . . . . . 16 |- ( ( V = { A , B , C } /\ G e. USGraph ) -> ( { { B , A } , { B , B } } C_ E -> B = A ) )
84833ad2ant3 1084 . . . . . . . . . . . . . . 15 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> ( { { B , A } , { B , B } } C_ E -> B = A ) )
8584com12 32 . . . . . . . . . . . . . 14 |- ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = A ) )
86852a1i 12 . . . . . . . . . . . . 13 |- ( y = A -> ( { { A , A } , { A , B } } C_ E -> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = A ) ) ) )
87 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = A -> ( B = y <-> B = A ) )
8887imbi2d 330 . . . . . . . . . . . . . 14 |- ( y = A -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = A ) ) )
8988imbi2d 330 . . . . . . . . . . . . 13 |- ( y = A -> ( ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) <-> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = A ) ) ) )
9086, 11, 893imtr4d 283 . . . . . . . . . . . 12 |- ( y = A -> ( { { y , A } , { y , B } } C_ E -> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) ) )
91 eqidd 2623 . . . . . . . . . . . . . . 15 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = B )
9291a1i 11 . . . . . . . . . . . . . 14 |- ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = B ) )
9392a1i13 27 . . . . . . . . . . . . 13 |- ( y = B -> ( { { B , A } , { B , B } } C_ E -> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = B ) ) ) )
94 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = B -> ( B = y <-> B = B ) )
9594imbi2d 330 . . . . . . . . . . . . . 14 |- ( y = B -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = B ) ) )
9695imbi2d 330 . . . . . . . . . . . . 13 |- ( y = B -> ( ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) <-> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = B ) ) ) )
9793, 37, 963imtr4d 283 . . . . . . . . . . . 12 |- ( y = B -> ( { { y , A } , { y , B } } C_ E -> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) ) )
9876pm2.24i 146 . . . . . . . . . . . . . . . . . . . . . 22 |- ( -. B = B -> B = C )
9975, 98sylbi 207 . . . . . . . . . . . . . . . . . . . . 21 |- ( B =/= B -> B = C )
10074, 99syl 17 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( V = { A , B , C } /\ G e. USGraph ) /\ { B , B } e. E ) -> B = C )
101100expcom 451 . . . . . . . . . . . . . . . . . . 19 |- ( { B , B } e. E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> B = C ) )
102101adantl 482 . . . . . . . . . . . . . . . . . 18 |- ( ( { B , A } e. E /\ { B , B } e. E ) -> ( ( V = { A , B , C } /\ G e. USGraph ) -> B = C ) )
10372, 102sylbir 225 . . . . . . . . . . . . . . . . 17 |- ( { { B , A } , { B , B } } C_ E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> B = C ) )
104103com12 32 . . . . . . . . . . . . . . . 16 |- ( ( V = { A , B , C } /\ G e. USGraph ) -> ( { { B , A } , { B , B } } C_ E -> B = C ) )
1051043ad2ant3 1084 . . . . . . . . . . . . . . 15 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> ( { { B , A } , { B , B } } C_ E -> B = C ) )
106105com12 32 . . . . . . . . . . . . . 14 |- ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = C ) )
1071062a1i 12 . . . . . . . . . . . . 13 |- ( y = C -> ( { { C , A } , { C , B } } C_ E -> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = C ) ) ) )
108 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = C -> ( B = y <-> B = C ) )
109108imbi2d 330 . . . . . . . . . . . . . 14 |- ( y = C -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = C ) ) )
110109imbi2d 330 . . . . . . . . . . . . 13 |- ( y = C -> ( ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) <-> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = C ) ) ) )
111107, 55, 1103imtr4d 283 . . . . . . . . . . . 12 |- ( y = C -> ( { { y , A } , { y , B } } C_ E -> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) ) )
11290, 97, 1113jaoi 1391 . . . . . . . . . . 11 |- ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ E -> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) ) )
113 preq1 4268 . . . . . . . . . . . . . . 15 |- ( x = B -> { x , A } = { B , A } )
114 preq1 4268 . . . . . . . . . . . . . . 15 |- ( x = B -> { x , B } = { B , B } )
115113, 114preq12d 4276 . . . . . . . . . . . . . 14 |- ( x = B -> { { x , A } , { x , B } } = { { B , A } , { B , B } } )
116115sseq1d 3632 . . . . . . . . . . . . 13 |- ( x = B -> ( { { x , A } , { x , B } } C_ E <-> { { B , A } , { B , B } } C_ E ) )
117 eqeq1 2626 . . . . . . . . . . . . . 14 |- ( x = B -> ( x = y <-> B = y ) )
118117imbi2d 330 . . . . . . . . . . . . 13 |- ( x = B -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) )
119116, 118imbi12d 334 . . . . . . . . . . . 12 |- ( x = B -> ( ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) <-> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) ) )
120119imbi2d 330 . . . . . . . . . . 11 |- ( x = B -> ( ( { { y , A } , { y , B } } C_ E -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) <-> ( { { y , A } , { y , B } } C_ E -> ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> B = y ) ) ) ) )
121112, 120syl5ibr 236 . . . . . . . . . 10 |- ( x = B -> ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ E -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) ) )
12223pm2.24i 146 . . . . . . . . . . . . . . . . . . . . . 22 |- ( -. A = A -> C = A )
12322, 122sylbi 207 . . . . . . . . . . . . . . . . . . . . 21 |- ( A =/= A -> C = A )
12421, 123syl 17 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( V = { A , B , C } /\ G e. USGraph ) /\ { A , A } e. E ) -> C = A )
125124expcom 451 . . . . . . . . . . . . . . . . . . 19 |- ( { A , A } e. E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> C = A ) )
126125adantr 481 . . . . . . . . . . . . . . . . . 18 |- ( ( { A , A } e. E /\ { A , B } e. E ) -> ( ( V = { A , B , C } /\ G e. USGraph ) -> C = A ) )
12718, 126sylbir 225 . . . . . . . . . . . . . . . . 17 |- ( { { A , A } , { A , B } } C_ E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> C = A ) )
128127com12 32 . . . . . . . . . . . . . . . 16 |- ( ( V = { A , B , C } /\ G e. USGraph ) -> ( { { A , A } , { A , B } } C_ E -> C = A ) )
1291283ad2ant3 1084 . . . . . . . . . . . . . . 15 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> ( { { A , A } , { A , B } } C_ E -> C = A ) )
130129com12 32 . . . . . . . . . . . . . 14 |- ( { { A , A } , { A , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = A ) )
131130a1i13 27 . . . . . . . . . . . . 13 |- ( y = A -> ( { { A , A } , { A , B } } C_ E -> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = A ) ) ) )
132 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = A -> ( C = y <-> C = A ) )
133132imbi2d 330 . . . . . . . . . . . . . 14 |- ( y = A -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = A ) ) )
134133imbi2d 330 . . . . . . . . . . . . 13 |- ( y = A -> ( ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) <-> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = A ) ) ) )
135131, 11, 1343imtr4d 283 . . . . . . . . . . . 12 |- ( y = A -> ( { { y , A } , { y , B } } C_ E -> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) ) )
136 pm2.21 120 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( -. B = B -> ( B = B -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
13775, 136sylbi 207 . . . . . . . . . . . . . . . . . . . . . 22 |- ( B =/= B -> ( B = B -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
13874, 76, 137mpisyl 21 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( V = { A , B , C } /\ G e. USGraph ) /\ { B , B } e. E ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) )
139138expcom 451 . . . . . . . . . . . . . . . . . . . 20 |- ( { B , B } e. E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
140139adantl 482 . . . . . . . . . . . . . . . . . . 19 |- ( ( { B , A } e. E /\ { B , B } e. E ) -> ( ( V = { A , B , C } /\ G e. USGraph ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
14172, 140sylbir 225 . . . . . . . . . . . . . . . . . 18 |- ( { { B , A } , { B , B } } C_ E -> ( ( V = { A , B , C } /\ G e. USGraph ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
142141com13 88 . . . . . . . . . . . . . . . . 17 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( V = { A , B , C } /\ G e. USGraph ) -> ( { { B , A } , { B , B } } C_ E -> C = B ) ) )
143142a1d 25 . . . . . . . . . . . . . . . 16 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( ( V = { A , B , C } /\ G e. USGraph ) -> ( { { B , A } , { B , B } } C_ E -> C = B ) ) ) )
1441433imp 1256 . . . . . . . . . . . . . . 15 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> ( { { B , A } , { B , B } } C_ E -> C = B ) )
145144com12 32 . . . . . . . . . . . . . 14 |- ( { { B , A } , { B , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = B ) )
146145a1i13 27 . . . . . . . . . . . . 13 |- ( y = B -> ( { { B , A } , { B , B } } C_ E -> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = B ) ) ) )
147 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = B -> ( C = y <-> C = B ) )
148147imbi2d 330 . . . . . . . . . . . . . 14 |- ( y = B -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = B ) ) )
149148imbi2d 330 . . . . . . . . . . . . 13 |- ( y = B -> ( ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) <-> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = B ) ) ) )
150146, 37, 1493imtr4d 283 . . . . . . . . . . . 12 |- ( y = B -> ( { { y , A } , { y , B } } C_ E -> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) ) )
151 eqidd 2623 . . . . . . . . . . . . . . 15 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = C )
152151a1i 11 . . . . . . . . . . . . . 14 |- ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = C ) )
153152a1i13 27 . . . . . . . . . . . . 13 |- ( y = C -> ( { { C , A } , { C , B } } C_ E -> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = C ) ) ) )
154 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = C -> ( C = y <-> C = C ) )
155154imbi2d 330 . . . . . . . . . . . . . 14 |- ( y = C -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = C ) ) )
156155imbi2d 330 . . . . . . . . . . . . 13 |- ( y = C -> ( ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) <-> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = C ) ) ) )
157153, 55, 1563imtr4d 283 . . . . . . . . . . . 12 |- ( y = C -> ( { { y , A } , { y , B } } C_ E -> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) ) )
158135, 150, 1573jaoi 1391 . . . . . . . . . . 11 |- ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ E -> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) ) )
159 preq1 4268 . . . . . . . . . . . . . . 15 |- ( x = C -> { x , A } = { C , A } )
160 preq1 4268 . . . . . . . . . . . . . . 15 |- ( x = C -> { x , B } = { C , B } )
161159, 160preq12d 4276 . . . . . . . . . . . . . 14 |- ( x = C -> { { x , A } , { x , B } } = { { C , A } , { C , B } } )
162161sseq1d 3632 . . . . . . . . . . . . 13 |- ( x = C -> ( { { x , A } , { x , B } } C_ E <-> { { C , A } , { C , B } } C_ E ) )
163 eqeq1 2626 . . . . . . . . . . . . . 14 |- ( x = C -> ( x = y <-> C = y ) )
164163imbi2d 330 . . . . . . . . . . . . 13 |- ( x = C -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) <-> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) )
165162, 164imbi12d 334 . . . . . . . . . . . 12 |- ( x = C -> ( ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) <-> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) ) )
166165imbi2d 330 . . . . . . . . . . 11 |- ( x = C -> ( ( { { y , A } , { y , B } } C_ E -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) <-> ( { { y , A } , { y , B } } C_ E -> ( { { C , A } , { C , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> C = y ) ) ) ) )
167158, 166syl5ibr 236 . . . . . . . . . 10 |- ( x = C -> ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ E -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) ) )
16869, 121, 1673jaoi 1391 . . . . . . . . 9 |- ( ( x = A \/ x = B \/ x = C ) -> ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ E -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) ) )
169168com3l 89 . . . . . . . 8 |- ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ E -> ( ( x = A \/ x = B \/ x = C ) -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) ) )
1704, 169sylbi 207 . . . . . . 7 |- ( y e. { A , B , C } -> ( { { y , A } , { y , B } } C_ E -> ( ( x = A \/ x = B \/ x = C ) -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) ) )
171170imp 445 . . . . . 6 |- ( ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ E ) -> ( ( x = A \/ x = B \/ x = C ) -> ( { { x , A } , { x , B } } C_ E -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) )
172171com3l 89 . . . . 5 |- ( ( x = A \/ x = B \/ x = C ) -> ( { { x , A } , { x , B } } C_ E -> ( ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ E ) -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) )
1732, 172sylbi 207 . . . 4 |- ( x e. { A , B , C } -> ( { { x , A } , { x , B } } C_ E -> ( ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ E ) -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) ) ) )
174173imp31 448 . . 3 |- ( ( ( x e. { A , B , C } /\ { { x , A } , { x , B } } C_ E ) /\ ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ E ) ) -> ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> x = y ) )
175174com12 32 . 2 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> ( ( ( x e. { A , B , C } /\ { { x , A } , { x , B } } C_ E ) /\ ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ E ) ) -> x = y ) )
176175alrimivv 1856 1 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( V = { A , B , C } /\ G e. USGraph ) ) -> A. x A. y ( ( ( x e. { A , B , C } /\ { { x , A } , { x , B } } C_ E ) /\ ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ E ) ) -> x = y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   \/ w3o 1036   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   {cpr 4179   {ctp 4181   ` cfv 5888  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044
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-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-oadd 7564  df-er 7742  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-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-umgr 25978  df-usgr 26046
This theorem is referenced by:  frgr3vlem2  27138
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