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Theorem 3vfriswmgralem 24836
Description: Lemma for 3vfriswmgra 24837. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
3vfriswmgralem  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  B }  e.  ran  E  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E ) )
Distinct variable groups:    w, A    w, B    w, C    w, E    w, X    w, Y

Proof of Theorem 3vfriswmgralem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . . 7  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  { A ,  B }  e.  ran  E )
21olcd 393 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) )
3 preq2 4113 . . . . . . . . . 10  |-  ( w  =  A  ->  { A ,  w }  =  { A ,  A }
)
43eleq1d 2536 . . . . . . . . 9  |-  ( w  =  A  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  A }  e.  ran  E ) )
5 preq2 4113 . . . . . . . . . 10  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
65eleq1d 2536 . . . . . . . . 9  |-  ( w  =  B  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
74, 6rexprg 4083 . . . . . . . 8  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( E. w  e. 
{ A ,  B }  { A ,  w }  e.  ran  E  <->  ( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) ) )
873ad2ant1 1017 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) ) )
98adantr 465 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) ) )
102, 9mpbird 232 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E )
11 df-rex 2823 . . . . 5  |-  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <->  E. w ( w  e. 
{ A ,  B }  /\  { A ,  w }  e.  ran  E ) )
1210, 11sylib 196 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E. w
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
13 vex 3121 . . . . . . . . 9  |-  w  e. 
_V
1413elpr 4051 . . . . . . . 8  |-  ( w  e.  { A ,  B }  <->  ( w  =  A  \/  w  =  B ) )
15 vex 3121 . . . . . . . . . . . 12  |-  y  e. 
_V
1615elpr 4051 . . . . . . . . . . 11  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
17 eqidd 2468 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A )
1817a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) )
1918a1ii 27 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  A }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) ) ) )
20 preq2 4113 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  { A ,  y }  =  { A ,  A }
)
2120eleq1d 2536 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  <->  { A ,  A }  e.  ran  E ) )
22 eqeq2 2482 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  A  ->  ( A  =  y  <->  A  =  A ) )
2322imbi2d 316 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) ) )
2423imbi2d 316 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  (
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) )  <->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) ) ) )
2519, 21, 243imtr4d 268 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
26 usgraedgrn 24213 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
27 df-ne 2664 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  A  <->  -.  A  =  A )
28 eqid 2467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  A  =  A
2928pm2.24i 144 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  A  =  A  ->  A  =  B )
3027, 29sylbi 195 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  =/=  A  ->  A  =  B )
3126, 30syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =  B )
3231ex 434 . . . . . . . . . . . . . . . . . . . 20  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
33323ad2ant3 1019 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
3433adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
3534com12 31 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) )
3635a1ii 27 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  B }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) ) ) )
37 preq2 4113 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
3837eleq1d 2536 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
39 eqeq2 2482 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
4039imbi2d 316 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) ) )
4140imbi2d 316 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  (
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) )  <->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) ) ) )
4236, 38, 413imtr4d 268 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
4325, 42jaoi 379 . . . . . . . . . . . . . 14  |-  ( ( y  =  A  \/  y  =  B )  ->  ( { A , 
y }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
44 eqeq1 2471 . . . . . . . . . . . . . . . . 17  |-  ( w  =  A  ->  (
w  =  y  <->  A  =  y ) )
4544imbi2d 316 . . . . . . . . . . . . . . . 16  |-  ( w  =  A  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) )
464, 45imbi12d 320 . . . . . . . . . . . . . . 15  |-  ( w  =  A  ->  (
( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) )  <->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
4746imbi2d 316 . . . . . . . . . . . . . 14  |-  ( w  =  A  ->  (
( { A , 
y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) )  <-> 
( { A , 
y }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) ) )
4843, 47syl5ibr 221 . . . . . . . . . . . . 13  |-  ( w  =  A  ->  (
( y  =  A  \/  y  =  B )  ->  ( { A ,  y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
4928pm2.24i 144 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  A  =  A  ->  B  =  A )
5027, 49sylbi 195 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  A  ->  B  =  A )
5126, 50syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  B  =  A )
5251ex 434 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
53523ad2ant3 1019 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
5453adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
5554com12 31 . . . . . . . . . . . . . . . . . 18  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) )
5655a1d 25 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) )
5756a1i 11 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) ) )
58 eqeq2 2482 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  A  ->  ( B  =  y  <->  B  =  A ) )
5958imbi2d 316 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) )
6059imbi2d 316 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  (
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) )  <->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) ) )
6157, 21, 603imtr4d 268 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
62 eqidd 2468 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B )
6362a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) )
6463a1ii 27 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  B }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) ) ) )
65 eqeq2 2482 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  B  ->  ( B  =  y  <->  B  =  B ) )
6665imbi2d 316 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) ) )
6766imbi2d 316 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  (
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) )  <->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) ) ) )
6864, 38, 673imtr4d 268 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
6961, 68jaoi 379 . . . . . . . . . . . . . 14  |-  ( ( y  =  A  \/  y  =  B )  ->  ( { A , 
y }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
70 eqeq1 2471 . . . . . . . . . . . . . . . . 17  |-  ( w  =  B  ->  (
w  =  y  <->  B  =  y ) )
7170imbi2d 316 . . . . . . . . . . . . . . . 16  |-  ( w  =  B  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) )
726, 71imbi12d 320 . . . . . . . . . . . . . . 15  |-  ( w  =  B  ->  (
( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) )  <->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
7372imbi2d 316 . . . . . . . . . . . . . 14  |-  ( w  =  B  ->  (
( { A , 
y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) )  <-> 
( { A , 
y }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) ) )
7469, 73syl5ibr 221 . . . . . . . . . . . . 13  |-  ( w  =  B  ->  (
( y  =  A  \/  y  =  B )  ->  ( { A ,  y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7548, 74jaoi 379 . . . . . . . . . . . 12  |-  ( ( w  =  A  \/  w  =  B )  ->  ( ( y  =  A  \/  y  =  B )  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7675com3l 81 . . . . . . . . . . 11  |-  ( ( y  =  A  \/  y  =  B )  ->  ( { A , 
y }  e.  ran  E  ->  ( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7716, 76sylbi 195 . . . . . . . . . 10  |-  ( y  e.  { A ,  B }  ->  ( { A ,  y }  e.  ran  E  -> 
( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7877imp 429 . . . . . . . . 9  |-  ( ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E )  -> 
( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) )
7978com3l 81 . . . . . . . 8  |-  ( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  ( ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E )  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) )
8014, 79sylbi 195 . . . . . . 7  |-  ( w  e.  { A ,  B }  ->  ( { A ,  w }  e.  ran  E  ->  (
( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
)  ->  ( (
( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) )
8180imp31 432 . . . . . 6  |-  ( ( ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e. 
{ A ,  B }  /\  { A , 
y }  e.  ran  E ) )  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) )
8281com12 31 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( (
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e. 
{ A ,  B }  /\  { A , 
y }  e.  ran  E ) )  ->  w  =  y ) )
8382alrimivv 1696 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A. w A. y ( ( ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) )  ->  w  =  y ) )
84 eleq1 2539 . . . . . 6  |-  ( w  =  y  ->  (
w  e.  { A ,  B }  <->  y  e.  { A ,  B }
) )
85 preq2 4113 . . . . . . 7  |-  ( w  =  y  ->  { A ,  w }  =  { A ,  y }
)
8685eleq1d 2536 . . . . . 6  |-  ( w  =  y  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  y }  e.  ran  E ) )
8784, 86anbi12d 710 . . . . 5  |-  ( w  =  y  ->  (
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  <-> 
( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) ) )
8887eu4 2340 . . . 4  |-  ( E! w ( w  e. 
{ A ,  B }  /\  { A ,  w }  e.  ran  E )  <->  ( E. w
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  A. w A. y ( ( ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) )  ->  w  =  y ) ) )
8912, 83, 88sylanbrc 664 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E! w
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
90 df-reu 2824 . . 3  |-  ( E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
E! w ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
9189, 90sylibr 212 . 2  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E )
9291ex 434 1  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  B }  e.  ran  E  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275    =/= wne 2662   E.wrex 2818   E!wreu 2819   {cpr 4035   {ctp 4037   class class class wbr 4453   ran crn 5006   USGrph cusg 24162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386  df-usgra 24165
This theorem is referenced by:  3vfriswmgra  24837
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