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Theorem frgra3vlem1 30589
Description: Lemma 1 for frgra3v 30591. (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 2973 . . . . . 6  |-  x  e. 
_V
21eltp 3919 . . . . 5  |-  ( x  e.  { A ,  B ,  C }  <->  ( x  =  A  \/  x  =  B  \/  x  =  C )
)
3 vex 2973 . . . . . . . . 9  |-  y  e. 
_V
43eltp 3919 . . . . . . . 8  |-  ( y  e.  { A ,  B ,  C }  <->  ( y  =  A  \/  y  =  B  \/  y  =  C )
)
5 eqidd 2442 . . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  { y ,  A }  =  { A ,  A }
)
9 preq1 3952 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  { y ,  B }  =  { A ,  B }
)
108, 9preq12d 3960 . . . . . . . . . . . . . 14  |-  ( y  =  A  ->  { {
y ,  A } ,  { y ,  B } }  =  { { A ,  A } ,  { A ,  B } } )
1110sseq1d 3381 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  ( { { y ,  A } ,  { y ,  B } }  C_  ran  E  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E ) )
12 eqeq2 2450 . . . . . . . . . . . . . . 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 4532 . . . . . . . . . . . . . . . . 17  |-  { A ,  A }  e.  _V
17 prex 4532 . . . . . . . . . . . . . . . . 17  |-  { A ,  B }  e.  _V
1816, 17prss 4025 . . . . . . . . . . . . . . . 16  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E )
19 usgraedgrn 23298 . . . . . . . . . . . . . . . . . . 19  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
20 df-ne 2606 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  =/=  A  <->  -.  A  =  A )
21 eqid 2441 . . . . . . . . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  { y ,  A }  =  { B ,  A }
)
31 preq1 3952 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  { y ,  B }  =  { B ,  B }
)
3230, 31preq12d 3960 . . . . . . . . . . . . . 14  |-  ( y  =  B  ->  { {
y ,  A } ,  { y ,  B } }  =  { { B ,  A } ,  { B ,  B } } )
3332sseq1d 3381 . . . . . . . . . . . . 13  |-  ( y  =  B  ->  ( { { y ,  A } ,  { y ,  B } }  C_  ran  E  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E ) )
34 eqeq2 2450 . . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . 15  |-  ( y  =  C  ->  { y ,  A }  =  { C ,  A }
)
47 preq1 3952 . . . . . . . . . . . . . . 15  |-  ( y  =  C  ->  { y ,  B }  =  { C ,  B }
)
4846, 47preq12d 3960 . . . . . . . . . . . . . 14  |-  ( y  =  C  ->  { {
y ,  A } ,  { y ,  B } }  =  { { C ,  A } ,  { C ,  B } } )
4948sseq1d 3381 . . . . . . . . . . . . 13  |-  ( y  =  C  ->  ( { { y ,  A } ,  { y ,  B } }  C_  ran  E  <->  { { C ,  A } ,  { C ,  B } }  C_  ran  E ) )
50 eqeq2 2450 . . . . . . . . . . . . . . 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 1281 . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  { x ,  A }  =  { A ,  A }
)
56 preq1 3952 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
5755, 56preq12d 3960 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { {
x ,  A } ,  { x ,  B } }  =  { { A ,  A } ,  { A ,  B } } )
5857sseq1d 3381 . . . . . . . . . . . . 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 4532 . . . . . . . . . . . . . . . . 17  |-  { B ,  A }  e.  _V
65 prex 4532 . . . . . . . . . . . . . . . . 17  |-  { B ,  B }  e.  _V
6664, 65prss 4025 . . . . . . . . . . . . . . . 16  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  B }  e.  ran  E )  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E )
67 usgraedgrn 23298 . . . . . . . . . . . . . . . . . . 19  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  B  =/=  B )
68 df-ne 2606 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =/=  B  <->  -.  B  =  B )
69 eqid 2441 . . . . . . . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . 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 2442 . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . 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 1281 . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . 15  |-  ( x  =  B  ->  { x ,  A }  =  { B ,  A }
)
103 preq1 3952 . . . . . . . . . . . . . . 15  |-  ( x  =  B  ->  { x ,  B }  =  { B ,  B }
)
104102, 103preq12d 3960 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { {
x ,  A } ,  { x ,  B } }  =  { { B ,  A } ,  { B ,  B } } )
105104sseq1d 3381 . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . 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 2442 . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . 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 1281 . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  { x ,  A }  =  { C ,  A }
)
149 preq1 3952 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  { x ,  B }  =  { C ,  B }
)
150148, 149preq12d 3960 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { {
x ,  A } ,  { x ,  B } }  =  { { C ,  A } ,  { C ,  B } } )
151150sseq1d 3381 . . . . . . . . . . . . 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 1281 . . . . . . . . 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 1686 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 964    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756    =/= wne 2604    C_ wss 3326   {cpr 3877   {ctp 3879   class class class wbr 4290   ran crn 4839   USGrph cusg 23262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-hash 12102  df-usgra 23264
This theorem is referenced by:  frgra3vlem2  30590
  Copyright terms: Public domain W3C validator