MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgra3vlem1 Structured version   Unicode version
Theorem frgra3vlem1 24978
Description: Lemma 1 for frgra3v 24980. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frgra3vlem1 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A. x A. y ( ( ( x e. { A , B , C } /\ { { x , A } , { x , B } } C_ ran E ) /\ ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ ran E ) ) -> x = y ) )
Distinct variable groups:   x, A, y   x, B, y   x, C, y   x, E, y   x, X, y   x, Y, y   x, Z, y
Proof of Theorem frgra3vlem1
StepHypRef Expression
1 vex 3098 . . . . . 6 |- x e. _V
21eltp 4059 . . . . 5 |- ( x e. { A , B , C } <-> ( x = A \/ x = B \/ x = C ) )
3 vex 3098 . . . . . . . . 9 |- y e. _V
43eltp 4059 . . . . . . . 8 |- ( y e. { A , B , C } <-> ( y = A \/ y = B \/ y = C ) )
5 eqidd 2444 . . . . . . . . . . . . . . 15 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = A )
65a1i 11 . . . . . . . . . . . . . 14 |- ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = A ) )
76a1ii 27 . . . . . . . . . . . . 13 |- ( y = A -> ( { { A , A } , { A , B } } C_ ran E -> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = A ) ) ) )
8 preq1 4094 . . . . . . . . . . . . . . 15 |- ( y = A -> { y , A } = { A , A } )
9 preq1 4094 . . . . . . . . . . . . . . 15 |- ( y = A -> { y , B } = { A , B } )
108, 9preq12d 4102 . . . . . . . . . . . . . 14 |- ( y = A -> { { y , A } , { y , B } } = { { A , A } , { A , B } } )
1110sseq1d 3516 . . . . . . . . . . . . 13 |- ( y = A -> ( { { y , A } , { y , B } } C_ ran E <-> { { A , A } , { A , B } } C_ ran E ) )
12 eqeq2 2458 . . . . . . . . . . . . . . 15 |- ( y = A -> ( A = y <-> A = A ) )
1312imbi2d 316 . . . . . . . . . . . . . 14 |- ( y = A -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = A ) ) )
1413imbi2d 316 . . . . . . . . . . . . 13 |- ( y = A -> ( ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) <-> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = A ) ) ) )
157, 11, 143imtr4d 268 . . . . . . . . . . . 12 |- ( y = A -> ( { { y , A } , { y , B } } C_ ran E -> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) ) )
16 prex 4679 . . . . . . . . . . . . . . . . 17 |- { A , A } e. _V
17 prex 4679 . . . . . . . . . . . . . . . . 17 |- { A , B } e. _V
1816, 17prss 4169 . . . . . . . . . . . . . . . 16 |- ( ( { A , A } e. ran E /\ { A , B } e. ran E ) <-> { { A , A } , { A , B } } C_ ran E )
19 usgraedgrn 24359 . . . . . . . . . . . . . . . . . . 19 |- ( ( { A , B , C } USGrph E /\ { A , A } e. ran E ) -> A =/= A )
20 df-ne 2640 . . . . . . . . . . . . . . . . . . . 20 |- ( A =/= A <-> -. A = A )
21 eqid 2443 . . . . . . . . . . . . . . . . . . . . 21 |- A = A
2221pm2.24i 144 . . . . . . . . . . . . . . . . . . . 20 |- ( -. A = A -> A = B )
2320, 22sylbi 195 . . . . . . . . . . . . . . . . . . 19 |- ( A =/= A -> A = B )
2419, 23syl 16 . . . . . . . . . . . . . . . . . 18 |- ( ( { A , B , C } USGrph E /\ { A , A } e. ran E ) -> A = B )
2524expcom 435 . . . . . . . . . . . . . . . . 17 |- ( { A , A } e. ran E -> ( { A , B , C } USGrph E -> A = B ) )
2625adantr 465 . . . . . . . . . . . . . . . 16 |- ( ( { A , A } e. ran E /\ { A , B } e. ran E ) -> ( { A , B , C } USGrph E -> A = B ) )
2718, 26sylbir 213 . . . . . . . . . . . . . . 15 |- ( { { A , A } , { A , B } } C_ ran E -> ( { A , B , C } USGrph E -> A = B ) )
2827adantld 467 . . . . . . . . . . . . . 14 |- ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = B ) )
2928a1ii 27 . . . . . . . . . . . . 13 |- ( y = B -> ( { { B , A } , { B , B } } C_ ran E -> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = B ) ) ) )
30 preq1 4094 . . . . . . . . . . . . . . 15 |- ( y = B -> { y , A } = { B , A } )
31 preq1 4094 . . . . . . . . . . . . . . 15 |- ( y = B -> { y , B } = { B , B } )
3230, 31preq12d 4102 . . . . . . . . . . . . . 14 |- ( y = B -> { { y , A } , { y , B } } = { { B , A } , { B , B } } )
3332sseq1d 3516 . . . . . . . . . . . . 13 |- ( y = B -> ( { { y , A } , { y , B } } C_ ran E <-> { { B , A } , { B , B } } C_ ran E ) )
34 eqeq2 2458 . . . . . . . . . . . . . . 15 |- ( y = B -> ( A = y <-> A = B ) )
3534imbi2d 316 . . . . . . . . . . . . . 14 |- ( y = B -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = B ) ) )
3635imbi2d 316 . . . . . . . . . . . . 13 |- ( y = B -> ( ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) <-> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = B ) ) ) )
3729, 33, 363imtr4d 268 . . . . . . . . . . . 12 |- ( y = B -> ( { { y , A } , { y , B } } C_ ran E -> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) ) )
3821pm2.24i 144 . . . . . . . . . . . . . . . . . . . 20 |- ( -. A = A -> A = C )
3920, 38sylbi 195 . . . . . . . . . . . . . . . . . . 19 |- ( A =/= A -> A = C )
4019, 39syl 16 . . . . . . . . . . . . . . . . . 18 |- ( ( { A , B , C } USGrph E /\ { A , A } e. ran E ) -> A = C )
4140expcom 435 . . . . . . . . . . . . . . . . 17 |- ( { A , A } e. ran E -> ( { A , B , C } USGrph E -> A = C ) )
4241adantr 465 . . . . . . . . . . . . . . . 16 |- ( ( { A , A } e. ran E /\ { A , B } e. ran E ) -> ( { A , B , C } USGrph E -> A = C ) )
4318, 42sylbir 213 . . . . . . . . . . . . . . 15 |- ( { { A , A } , { A , B } } C_ ran E -> ( { A , B , C } USGrph E -> A = C ) )
4443adantld 467 . . . . . . . . . . . . . 14 |- ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = C ) )
4544a1ii 27 . . . . . . . . . . . . 13 |- ( y = C -> ( { { C , A } , { C , B } } C_ ran E -> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = C ) ) ) )
46 preq1 4094 . . . . . . . . . . . . . . 15 |- ( y = C -> { y , A } = { C , A } )
47 preq1 4094 . . . . . . . . . . . . . . 15 |- ( y = C -> { y , B } = { C , B } )
4846, 47preq12d 4102 . . . . . . . . . . . . . 14 |- ( y = C -> { { y , A } , { y , B } } = { { C , A } , { C , B } } )
4948sseq1d 3516 . . . . . . . . . . . . 13 |- ( y = C -> ( { { y , A } , { y , B } } C_ ran E <-> { { C , A } , { C , B } } C_ ran E ) )
50 eqeq2 2458 . . . . . . . . . . . . . . 15 |- ( y = C -> ( A = y <-> A = C ) )
5150imbi2d 316 . . . . . . . . . . . . . 14 |- ( y = C -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = C ) ) )
5251imbi2d 316 . . . . . . . . . . . . 13 |- ( y = C -> ( ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) <-> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = C ) ) ) )
5345, 49, 523imtr4d 268 . . . . . . . . . . . 12 |- ( y = C -> ( { { y , A } , { y , B } } C_ ran E -> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) ) )
5415, 37, 533jaoi 1292 . . . . . . . . . . 11 |- ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ ran E -> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) ) )
55 preq1 4094 . . . . . . . . . . . . . . 15 |- ( x = A -> { x , A } = { A , A } )
56 preq1 4094 . . . . . . . . . . . . . . 15 |- ( x = A -> { x , B } = { A , B } )
5755, 56preq12d 4102 . . . . . . . . . . . . . 14 |- ( x = A -> { { x , A } , { x , B } } = { { A , A } , { A , B } } )
5857sseq1d 3516 . . . . . . . . . . . . 13 |- ( x = A -> ( { { x , A } , { x , B } } C_ ran E <-> { { A , A } , { A , B } } C_ ran E ) )
59 eqeq1 2447 . . . . . . . . . . . . . 14 |- ( x = A -> ( x = y <-> A = y ) )
6059imbi2d 316 . . . . . . . . . . . . 13 |- ( x = A -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) )
6158, 60imbi12d 320 . . . . . . . . . . . 12 |- ( x = A -> ( ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) <-> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) ) )
6261imbi2d 316 . . . . . . . . . . 11 |- ( x = A -> ( ( { { y , A } , { y , B } } C_ ran E -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) <-> ( { { y , A } , { y , B } } C_ ran E -> ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A = y ) ) ) ) )
6354, 62syl5ibr 221 . . . . . . . . . 10 |- ( x = A -> ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ ran E -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) ) )
64 prex 4679 . . . . . . . . . . . . . . . . 17 |- { B , A } e. _V
65 prex 4679 . . . . . . . . . . . . . . . . 17 |- { B , B } e. _V
6664, 65prss 4169 . . . . . . . . . . . . . . . 16 |- ( ( { B , A } e. ran E /\ { B , B } e. ran E ) <-> { { B , A } , { B , B } } C_ ran E )
67 usgraedgrn 24359 . . . . . . . . . . . . . . . . . . 19 |- ( ( { A , B , C } USGrph E /\ { B , B } e. ran E ) -> B =/= B )
68 df-ne 2640 . . . . . . . . . . . . . . . . . . . 20 |- ( B =/= B <-> -. B = B )
69 eqid 2443 . . . . . . . . . . . . . . . . . . . . 21 |- B = B
7069pm2.24i 144 . . . . . . . . . . . . . . . . . . . 20 |- ( -. B = B -> B = A )
7168, 70sylbi 195 . . . . . . . . . . . . . . . . . . 19 |- ( B =/= B -> B = A )
7267, 71syl 16 . . . . . . . . . . . . . . . . . 18 |- ( ( { A , B , C } USGrph E /\ { B , B } e. ran E ) -> B = A )
7372expcom 435 . . . . . . . . . . . . . . . . 17 |- ( { B , B } e. ran E -> ( { A , B , C } USGrph E -> B = A ) )
7473adantl 466 . . . . . . . . . . . . . . . 16 |- ( ( { B , A } e. ran E /\ { B , B } e. ran E ) -> ( { A , B , C } USGrph E -> B = A ) )
7566, 74sylbir 213 . . . . . . . . . . . . . . 15 |- ( { { B , A } , { B , B } } C_ ran E -> ( { A , B , C } USGrph E -> B = A ) )
7675adantld 467 . . . . . . . . . . . . . 14 |- ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = A ) )
7776a1ii 27 . . . . . . . . . . . . 13 |- ( y = A -> ( { { A , A } , { A , B } } C_ ran E -> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = A ) ) ) )
78 eqeq2 2458 . . . . . . . . . . . . . . 15 |- ( y = A -> ( B = y <-> B = A ) )
7978imbi2d 316 . . . . . . . . . . . . . 14 |- ( y = A -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = A ) ) )
8079imbi2d 316 . . . . . . . . . . . . 13 |- ( y = A -> ( ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) <-> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = A ) ) ) )
8177, 11, 803imtr4d 268 . . . . . . . . . . . 12 |- ( y = A -> ( { { y , A } , { y , B } } C_ ran E -> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) ) )
82 eqidd 2444 . . . . . . . . . . . . . . 15 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = B )
8382a1i 11 . . . . . . . . . . . . . 14 |- ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = B ) )
8483a1ii 27 . . . . . . . . . . . . 13 |- ( y = B -> ( { { B , A } , { B , B } } C_ ran E -> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = B ) ) ) )
85 eqeq2 2458 . . . . . . . . . . . . . . 15 |- ( y = B -> ( B = y <-> B = B ) )
8685imbi2d 316 . . . . . . . . . . . . . 14 |- ( y = B -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = B ) ) )
8786imbi2d 316 . . . . . . . . . . . . 13 |- ( y = B -> ( ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) <-> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = B ) ) ) )
8884, 33, 873imtr4d 268 . . . . . . . . . . . 12 |- ( y = B -> ( { { y , A } , { y , B } } C_ ran E -> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) ) )
8969pm2.24i 144 . . . . . . . . . . . . . . . . . . . 20 |- ( -. B = B -> B = C )
9068, 89sylbi 195 . . . . . . . . . . . . . . . . . . 19 |- ( B =/= B -> B = C )
9167, 90syl 16 . . . . . . . . . . . . . . . . . 18 |- ( ( { A , B , C } USGrph E /\ { B , B } e. ran E ) -> B = C )
9291expcom 435 . . . . . . . . . . . . . . . . 17 |- ( { B , B } e. ran E -> ( { A , B , C } USGrph E -> B = C ) )
9392adantl 466 . . . . . . . . . . . . . . . 16 |- ( ( { B , A } e. ran E /\ { B , B } e. ran E ) -> ( { A , B , C } USGrph E -> B = C ) )
9466, 93sylbir 213 . . . . . . . . . . . . . . 15 |- ( { { B , A } , { B , B } } C_ ran E -> ( { A , B , C } USGrph E -> B = C ) )
9594adantld 467 . . . . . . . . . . . . . 14 |- ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = C ) )
9695a1ii 27 . . . . . . . . . . . . 13 |- ( y = C -> ( { { C , A } , { C , B } } C_ ran E -> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = C ) ) ) )
97 eqeq2 2458 . . . . . . . . . . . . . . 15 |- ( y = C -> ( B = y <-> B = C ) )
9897imbi2d 316 . . . . . . . . . . . . . 14 |- ( y = C -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = C ) ) )
9998imbi2d 316 . . . . . . . . . . . . 13 |- ( y = C -> ( ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) <-> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = C ) ) ) )
10096, 49, 993imtr4d 268 . . . . . . . . . . . 12 |- ( y = C -> ( { { y , A } , { y , B } } C_ ran E -> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) ) )
10181, 88, 1003jaoi 1292 . . . . . . . . . . 11 |- ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ ran E -> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) ) )
102 preq1 4094 . . . . . . . . . . . . . . 15 |- ( x = B -> { x , A } = { B , A } )
103 preq1 4094 . . . . . . . . . . . . . . 15 |- ( x = B -> { x , B } = { B , B } )
104102, 103preq12d 4102 . . . . . . . . . . . . . 14 |- ( x = B -> { { x , A } , { x , B } } = { { B , A } , { B , B } } )
105104sseq1d 3516 . . . . . . . . . . . . 13 |- ( x = B -> ( { { x , A } , { x , B } } C_ ran E <-> { { B , A } , { B , B } } C_ ran E ) )
106 eqeq1 2447 . . . . . . . . . . . . . 14 |- ( x = B -> ( x = y <-> B = y ) )
107106imbi2d 316 . . . . . . . . . . . . 13 |- ( x = B -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) )
108105, 107imbi12d 320 . . . . . . . . . . . 12 |- ( x = B -> ( ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) <-> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) ) )
109108imbi2d 316 . . . . . . . . . . 11 |- ( x = B -> ( ( { { y , A } , { y , B } } C_ ran E -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) <-> ( { { y , A } , { y , B } } C_ ran E -> ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> B = y ) ) ) ) )
110101, 109syl5ibr 221 . . . . . . . . . 10 |- ( x = B -> ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ ran E -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) ) )
11121pm2.24i 144 . . . . . . . . . . . . . . . . . . . . 21 |- ( -. A = A -> C = A )
11220, 111sylbi 195 . . . . . . . . . . . . . . . . . . . 20 |- ( A =/= A -> C = A )
11319, 112syl 16 . . . . . . . . . . . . . . . . . . 19 |- ( ( { A , B , C } USGrph E /\ { A , A } e. ran E ) -> C = A )
114113expcom 435 . . . . . . . . . . . . . . . . . 18 |- ( { A , A } e. ran E -> ( { A , B , C } USGrph E -> C = A ) )
115114adantr 465 . . . . . . . . . . . . . . . . 17 |- ( ( { A , A } e. ran E /\ { A , B } e. ran E ) -> ( { A , B , C } USGrph E -> C = A ) )
11618, 115sylbir 213 . . . . . . . . . . . . . . . 16 |- ( { { A , A } , { A , B } } C_ ran E -> ( { A , B , C } USGrph E -> C = A ) )
117116adantld 467 . . . . . . . . . . . . . . 15 |- ( { { A , A } , { A , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = A ) )
118117a1d 25 . . . . . . . . . . . . . 14 |- ( { { A , A } , { A , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = A ) ) )
119118a1i 11 . . . . . . . . . . . . 13 |- ( y = A -> ( { { A , A } , { A , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = A ) ) ) )
120 eqeq2 2458 . . . . . . . . . . . . . . 15 |- ( y = A -> ( C = y <-> C = A ) )
121120imbi2d 316 . . . . . . . . . . . . . 14 |- ( y = A -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = A ) ) )
122121imbi2d 316 . . . . . . . . . . . . 13 |- ( y = A -> ( ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) <-> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = A ) ) ) )
123119, 11, 1223imtr4d 268 . . . . . . . . . . . 12 |- ( y = A -> ( { { y , A } , { y , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) ) )
124 pm2.21 108 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( -. B = B -> ( B = B -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
12568, 124sylbi 195 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( B =/= B -> ( B = B -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
12667, 69, 125mpisyl 18 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( { A , B , C } USGrph E /\ { B , B } e. ran E ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) )
127126expcom 435 . . . . . . . . . . . . . . . . . . . . 21 |- ( { B , B } e. ran E -> ( { A , B , C } USGrph E -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
128127adantl 466 . . . . . . . . . . . . . . . . . . . 20 |- ( ( { B , A } e. ran E /\ { B , B } e. ran E ) -> ( { A , B , C } USGrph E -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
12966, 128sylbir 213 . . . . . . . . . . . . . . . . . . 19 |- ( { { B , A } , { B , B } } C_ ran E -> ( { A , B , C } USGrph E -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> C = B ) ) )
130129com13 80 . . . . . . . . . . . . . . . . . 18 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( { A , B , C } USGrph E -> ( { { B , A } , { B , B } } C_ ran E -> C = B ) ) )
131130adantr 465 . . . . . . . . . . . . . . . . 17 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( { A , B , C } USGrph E -> ( { { B , A } , { B , B } } C_ ran E -> C = B ) ) )
132131imp 429 . . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> ( { { B , A } , { B , B } } C_ ran E -> C = B ) )
133132com12 31 . . . . . . . . . . . . . . 15 |- ( { { B , A } , { B , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = B ) )
134133a1d 25 . . . . . . . . . . . . . 14 |- ( { { B , A } , { B , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = B ) ) )
135134a1i 11 . . . . . . . . . . . . 13 |- ( y = B -> ( { { B , A } , { B , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = B ) ) ) )
136 eqeq2 2458 . . . . . . . . . . . . . . 15 |- ( y = B -> ( C = y <-> C = B ) )
137136imbi2d 316 . . . . . . . . . . . . . 14 |- ( y = B -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = B ) ) )
138137imbi2d 316 . . . . . . . . . . . . 13 |- ( y = B -> ( ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) <-> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = B ) ) ) )
139135, 33, 1383imtr4d 268 . . . . . . . . . . . 12 |- ( y = B -> ( { { y , A } , { y , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) ) )
140 eqidd 2444 . . . . . . . . . . . . . . 15 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = C )
141140a1i 11 . . . . . . . . . . . . . 14 |- ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = C ) )
142141a1ii 27 . . . . . . . . . . . . 13 |- ( y = C -> ( { { C , A } , { C , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = C ) ) ) )
143 eqeq2 2458 . . . . . . . . . . . . . . 15 |- ( y = C -> ( C = y <-> C = C ) )
144143imbi2d 316 . . . . . . . . . . . . . 14 |- ( y = C -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = C ) ) )
145144imbi2d 316 . . . . . . . . . . . . 13 |- ( y = C -> ( ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) <-> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = C ) ) ) )
146142, 49, 1453imtr4d 268 . . . . . . . . . . . 12 |- ( y = C -> ( { { y , A } , { y , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) ) )
147123, 139, 1463jaoi 1292 . . . . . . . . . . 11 |- ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) ) )
148 preq1 4094 . . . . . . . . . . . . . . 15 |- ( x = C -> { x , A } = { C , A } )
149 preq1 4094 . . . . . . . . . . . . . . 15 |- ( x = C -> { x , B } = { C , B } )
150148, 149preq12d 4102 . . . . . . . . . . . . . 14 |- ( x = C -> { { x , A } , { x , B } } = { { C , A } , { C , B } } )
151150sseq1d 3516 . . . . . . . . . . . . 13 |- ( x = C -> ( { { x , A } , { x , B } } C_ ran E <-> { { C , A } , { C , B } } C_ ran E ) )
152 eqeq1 2447 . . . . . . . . . . . . . 14 |- ( x = C -> ( x = y <-> C = y ) )
153152imbi2d 316 . . . . . . . . . . . . 13 |- ( x = C -> ( ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) <-> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) )
154151, 153imbi12d 320 . . . . . . . . . . . 12 |- ( x = C -> ( ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) <-> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) ) )
155154imbi2d 316 . . . . . . . . . . 11 |- ( x = C -> ( ( { { y , A } , { y , B } } C_ ran E -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) <-> ( { { y , A } , { y , B } } C_ ran E -> ( { { C , A } , { C , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> C = y ) ) ) ) )
156147, 155syl5ibr 221 . . . . . . . . . 10 |- ( x = C -> ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ ran E -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) ) )
15763, 110, 1563jaoi 1292 . . . . . . . . 9 |- ( ( x = A \/ x = B \/ x = C ) -> ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ ran E -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) ) )
158157com3l 81 . . . . . . . 8 |- ( ( y = A \/ y = B \/ y = C ) -> ( { { y , A } , { y , B } } C_ ran E -> ( ( x = A \/ x = B \/ x = C ) -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) ) )
1594, 158sylbi 195 . . . . . . 7 |- ( y e. { A , B , C } -> ( { { y , A } , { y , B } } C_ ran E -> ( ( x = A \/ x = B \/ x = C ) -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) ) )
160159imp 429 . . . . . 6 |- ( ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ ran E ) -> ( ( x = A \/ x = B \/ x = C ) -> ( { { x , A } , { x , B } } C_ ran E -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) )
161160com3l 81 . . . . 5 |- ( ( x = A \/ x = B \/ x = C ) -> ( { { x , A } , { x , B } } C_ ran E -> ( ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ ran E ) -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) )
1622, 161sylbi 195 . . . 4 |- ( x e. { A , B , C } -> ( { { x , A } , { x , B } } C_ ran E -> ( ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ ran E ) -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) ) ) )
163162imp31 432 . . 3 |- ( ( ( x e. { A , B , C } /\ { { x , A } , { x , B } } C_ ran E ) /\ ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ ran E ) ) -> ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> x = y ) )
164163com12 31 . 2 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> ( ( ( x e. { A , B , C } /\ { { x , A } , { x , B } } C_ ran E ) /\ ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ ran E ) ) -> x = y ) )
165164alrimivv 1707 1 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ { A , B , C } USGrph E ) -> A. x A. y ( ( ( x e. { A , B , C } /\ { { x , A } , { x , B } } C_ ran E ) /\ ( y e. { A , B , C } /\ { { y , A } , { y , B } } C_ ran E ) ) -> x = y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   \/ w3o 973   /\ w3a 974   A.wal 1381   = wceq 1383   e. wcel 1804   =/= wne 2638   C_ wss 3461   {cpr 4016   {ctp 4018   class class class wbr 4437   ran crn 4990   USGrph cusg 24308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-hash 12388  df-usgra 24311
This theorem is referenced by:  frgra3vlem2  24979
  Copyright terms: Public domain W3C validator