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Theorem frgra3vlem1 28112
Description: Lemma 1 for frgra3v 28114. (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 2927 . . . . . 6  |-  x  e. 
_V
21eltp 3821 . . . . 5  |-  ( x  e.  { A ,  B ,  C }  <->  ( x  =  A  \/  x  =  B  \/  x  =  C )
)
3 vex 2927 . . . . . . . . 9  |-  y  e. 
_V
43eltp 3821 . . . . . . . 8  |-  ( y  e.  { A ,  B ,  C }  <->  ( y  =  A  \/  y  =  B  \/  y  =  C )
)
5 eqidd 2413 . . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . 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 3851 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  { y ,  A }  =  { A ,  A }
)
9 preq1 3851 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  { y ,  B }  =  { A ,  B }
)
108, 9preq12d 3859 . . . . . . . . . . . . . 14  |-  ( y  =  A  ->  { {
y ,  A } ,  { y ,  B } }  =  { { A ,  A } ,  { A ,  B } } )
1110sseq1d 3343 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  ( { { y ,  A } ,  { y ,  B } }  C_  ran  E  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E ) )
12 eqeq2 2421 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  ( A  =  y  <->  A  =  A ) )
1312imbi2d 308 . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 4374 . . . . . . . . . . . . . . . . 17  |-  { A ,  A }  e.  _V
17 prex 4374 . . . . . . . . . . . . . . . . 17  |-  { A ,  B }  e.  _V
1816, 17prss 3920 . . . . . . . . . . . . . . . 16  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E )
19 usgraedgrn 21362 . . . . . . . . . . . . . . . . . . 19  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
20 df-ne 2577 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  =/=  A  <->  -.  A  =  A )
21 eqid 2412 . . . . . . . . . . . . . . . . . . . . 21  |-  A  =  A
2221pm2.24i 138 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  A  =  A  ->  A  =  B )
2320, 22sylbi 188 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =/=  A  ->  A  =  B )
2419, 23syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =  B )
2524expcom 425 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  A  =  B ) )
2625adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  -> 
( { A ,  B ,  C } USGrph  E  ->  A  =  B ) )
2718, 26sylbir 205 . . . . . . . . . . . . . . 15  |-  ( { { A ,  A } ,  { A ,  B } }  C_  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  A  =  B ) )
2827adantld 454 . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . 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 3851 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  { y ,  A }  =  { B ,  A }
)
31 preq1 3851 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  { y ,  B }  =  { B ,  B }
)
3230, 31preq12d 3859 . . . . . . . . . . . . . 14  |-  ( y  =  B  ->  { {
y ,  A } ,  { y ,  B } }  =  { { B ,  A } ,  { B ,  B } } )
3332sseq1d 3343 . . . . . . . . . . . . 13  |-  ( y  =  B  ->  ( { { y ,  A } ,  { y ,  B } }  C_  ran  E  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E ) )
34 eqeq2 2421 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
3534imbi2d 308 . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 138 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  A  =  A  ->  A  =  C )
3920, 38sylbi 188 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =/=  A  ->  A  =  C )
4019, 39syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =  C )
4140expcom 425 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  A  =  C ) )
4241adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  -> 
( { A ,  B ,  C } USGrph  E  ->  A  =  C ) )
4318, 42sylbir 205 . . . . . . . . . . . . . . 15  |-  ( { { A ,  A } ,  { A ,  B } }  C_  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  A  =  C ) )
4443adantld 454 . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . 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 3851 . . . . . . . . . . . . . . 15  |-  ( y  =  C  ->  { y ,  A }  =  { C ,  A }
)
47 preq1 3851 . . . . . . . . . . . . . . 15  |-  ( y  =  C  ->  { y ,  B }  =  { C ,  B }
)
4846, 47preq12d 3859 . . . . . . . . . . . . . 14  |-  ( y  =  C  ->  { {
y ,  A } ,  { y ,  B } }  =  { { C ,  A } ,  { C ,  B } } )
4948sseq1d 3343 . . . . . . . . . . . . 13  |-  ( y  =  C  ->  ( { { y ,  A } ,  { y ,  B } }  C_  ran  E  <->  { { C ,  A } ,  { C ,  B } }  C_  ran  E ) )
50 eqeq2 2421 . . . . . . . . . . . . . . 15  |-  ( y  =  C  ->  ( A  =  y  <->  A  =  C ) )
5150imbi2d 308 . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 1247 . . . . . . . . . . 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 3851 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  { x ,  A }  =  { A ,  A }
)
56 preq1 3851 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
5755, 56preq12d 3859 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { {
x ,  A } ,  { x ,  B } }  =  { { A ,  A } ,  { A ,  B } } )
5857sseq1d 3343 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E ) )
59 eqeq1 2418 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
6059imbi2d 308 . . . . . . . . . . . . 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 312 . . . . . . . . . . . 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 308 . . . . . . . . . . 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 213 . . . . . . . . . 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 4374 . . . . . . . . . . . . . . . . 17  |-  { B ,  A }  e.  _V
65 prex 4374 . . . . . . . . . . . . . . . . 17  |-  { B ,  B }  e.  _V
6664, 65prss 3920 . . . . . . . . . . . . . . . 16  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  B }  e.  ran  E )  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E )
67 usgraedgrn 21362 . . . . . . . . . . . . . . . . . . 19  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  B  =/=  B )
68 df-ne 2577 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =/=  B  <->  -.  B  =  B )
69 eqid 2412 . . . . . . . . . . . . . . . . . . . . 21  |-  B  =  B
7069pm2.24i 138 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  B  =  B  ->  B  =  A )
7168, 70sylbi 188 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =/=  B  ->  B  =  A )
7267, 71syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  B  =  A )
7372expcom 425 . . . . . . . . . . . . . . . . 17  |-  ( { B ,  B }  e.  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  B  =  A ) )
7473adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  B }  e.  ran  E )  -> 
( { A ,  B ,  C } USGrph  E  ->  B  =  A ) )
7566, 74sylbir 205 . . . . . . . . . . . . . . 15  |-  ( { { B ,  A } ,  { B ,  B } }  C_  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  B  =  A ) )
7675adantld 454 . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  ( B  =  y  <->  B  =  A ) )
7978imbi2d 308 . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 2413 . . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  ( B  =  y  <->  B  =  B ) )
8685imbi2d 308 . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 138 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  B  =  B  ->  B  =  C )
9068, 89sylbi 188 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =/=  B  ->  B  =  C )
9167, 90syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  B  =  C )
9291expcom 425 . . . . . . . . . . . . . . . . 17  |-  ( { B ,  B }  e.  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  B  =  C ) )
9392adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  B }  e.  ran  E )  -> 
( { A ,  B ,  C } USGrph  E  ->  B  =  C ) )
9466, 93sylbir 205 . . . . . . . . . . . . . . 15  |-  ( { { B ,  A } ,  { B ,  B } }  C_  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  B  =  C ) )
9594adantld 454 . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . 15  |-  ( y  =  C  ->  ( B  =  y  <->  B  =  C ) )
9897imbi2d 308 . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 1247 . . . . . . . . . . 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 3851 . . . . . . . . . . . . . . 15  |-  ( x  =  B  ->  { x ,  A }  =  { B ,  A }
)
103 preq1 3851 . . . . . . . . . . . . . . 15  |-  ( x  =  B  ->  { x ,  B }  =  { B ,  B }
)
104102, 103preq12d 3859 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { {
x ,  A } ,  { x ,  B } }  =  { { B ,  A } ,  { B ,  B } } )
105104sseq1d 3343 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E ) )
106 eqeq1 2418 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  (
x  =  y  <->  B  =  y ) )
107106imbi2d 308 . . . . . . . . . . . . 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 312 . . . . . . . . . . . 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 308 . . . . . . . . . . 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 213 . . . . . . . . . 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 138 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  A  =  A  ->  C  =  A )
11220, 111sylbi 188 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  =/=  A  ->  C  =  A )
11319, 112syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  C  =  A )
114113expcom 425 . . . . . . . . . . . . . . . . . 18  |-  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  C  =  A ) )
115114adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  -> 
( { A ,  B ,  C } USGrph  E  ->  C  =  A ) )
11618, 115sylbir 205 . . . . . . . . . . . . . . . 16  |-  ( { { A ,  A } ,  { A ,  B } }  C_  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  C  =  A ) )
117116adantld 454 . . . . . . . . . . . . . . 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 23 . . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  ( C  =  y  <->  C  =  A ) )
121120imbi2d 308 . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 pm2.21 102 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  B  =  B  -> 
( B  =  B  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  =  B ) ) )
12568, 124sylbi 188 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( B  =/=  B  ->  ( B  =  B  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  C  =  B ) ) )
12667, 69, 125ee10 1382 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  -> 
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  =  B ) )
127126expcom 425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { B ,  B }  e.  ran  E  ->  ( { A ,  B ,  C } USGrph  E  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  C  =  B ) ) )
128127adantl 453 . . . . . . . . . . . . . . . . . . . 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 205 . . . . . . . . . . . . . . . . . . 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 76 . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . 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 29 . . . . . . . . . . . . . . 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 23 . . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  ( C  =  y  <->  C  =  B ) )
137136imbi2d 308 . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 2413 . . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . 15  |-  ( y  =  C  ->  ( C  =  y  <->  C  =  C ) )
144143imbi2d 308 . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 1247 . . . . . . . . . . 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 3851 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  { x ,  A }  =  { C ,  A }
)
149 preq1 3851 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  { x ,  B }  =  { C ,  B }
)
150148, 149preq12d 3859 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { {
x ,  A } ,  { x ,  B } }  =  { { C ,  A } ,  { C ,  B } } )
151150sseq1d 3343 . . . . . . . . . . . . 13  |-  ( x  =  C  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { C ,  A } ,  { C ,  B } }  C_  ran  E ) )
152 eqeq1 2418 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  (
x  =  y  <->  C  =  y ) )
153152imbi2d 308 . . . . . . . . . . . . 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 312 . . . . . . . . . . . 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 308 . . . . . . . . . . 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 213 . . . . . . . . . 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 1247 . . . . . . . . 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 77 . . . . . . . 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 188 . . . . . . 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 419 . . . . . 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 77 . . . . 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 188 . . . 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 422 . . 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 29 . 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 1639 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    \/ w3o 935    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1721    =/= wne 2575    C_ wss 3288   {cpr 3783   {ctp 3784   class class class wbr 4180   ran crn 4846   USGrph cusg 21326
This theorem is referenced by:  frgra3vlem2  28113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-hash 11582  df-usgra 21328
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