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Theorem frgranbnb 28347
Description: If two neighbors of a specific vertex have a common neighbor in a friendship graph, then this common neighbor must be the specific vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Hypotheses
Ref Expression
frgranbnb.x  |-  ( ph  ->  X  e.  V )
frgranbnb.nx  |-  D  =  ( <. V ,  E >. Neighbors  X )
frgranbnb.f  |-  ( ph  ->  V FriendGrph  E )
Assertion
Ref Expression
frgranbnb  |-  ( (
ph  /\  ( U  e.  D  /\  W  e.  D )  /\  U  =/=  W )  ->  (
( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X ) )

Proof of Theorem frgranbnb
StepHypRef Expression
1 frgranbnb.f . . . 4  |-  ( ph  ->  V FriendGrph  E )
2 frisusgra 28319 . . . 4  |-  ( V FriendGrph  E  ->  V USGrph  E )
31, 2syl 16 . . 3  |-  ( ph  ->  V USGrph  E )
4 frgranbnb.nx . . . . . . . . . 10  |-  D  =  ( <. V ,  E >. Neighbors  X )
54eleq2i 2499 . . . . . . . . 9  |-  ( U  e.  D  <->  U  e.  ( <. V ,  E >. Neighbors  X ) )
6 nbgraeledg 21434 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( U  e.  ( <. V ,  E >. Neighbors  X )  <->  { U ,  X }  e.  ran  E ) )
76biimpd 199 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( U  e.  ( <. V ,  E >. Neighbors  X )  ->  { U ,  X }  e.  ran  E ) )
85, 7syl5bi 209 . . . . . . . 8  |-  ( V USGrph  E  ->  ( U  e.  D  ->  { U ,  X }  e.  ran  E ) )
94eleq2i 2499 . . . . . . . . 9  |-  ( W  e.  D  <->  W  e.  ( <. V ,  E >. Neighbors  X ) )
10 nbgraeledg 21434 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( W  e.  ( <. V ,  E >. Neighbors  X )  <->  { W ,  X }  e.  ran  E ) )
1110biimpd 199 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( W  e.  ( <. V ,  E >. Neighbors  X )  ->  { W ,  X }  e.  ran  E ) )
129, 11syl5bi 209 . . . . . . . 8  |-  ( V USGrph  E  ->  ( W  e.  D  ->  { W ,  X }  e.  ran  E ) )
138, 12anim12d 547 . . . . . . 7  |-  ( V USGrph  E  ->  ( ( U  e.  D  /\  W  e.  D )  ->  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) ) )
1413imp 419 . . . . . 6  |-  ( ( V USGrph  E  /\  ( U  e.  D  /\  W  e.  D )
)  ->  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )
15 nbgraisvtx 21435 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( U  e.  ( <. V ,  E >. Neighbors  X )  ->  U  e.  V ) )
165, 15syl5bi 209 . . . . . . . 8  |-  ( V USGrph  E  ->  ( U  e.  D  ->  U  e.  V ) )
17 nbgraisvtx 21435 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( W  e.  ( <. V ,  E >. Neighbors  X )  ->  W  e.  V ) )
189, 17syl5bi 209 . . . . . . . 8  |-  ( V USGrph  E  ->  ( W  e.  D  ->  W  e.  V ) )
1916, 18anim12d 547 . . . . . . 7  |-  ( V USGrph  E  ->  ( ( U  e.  D  /\  W  e.  D )  ->  ( U  e.  V  /\  W  e.  V )
) )
2019imp 419 . . . . . 6  |-  ( ( V USGrph  E  /\  ( U  e.  D  /\  W  e.  D )
)  ->  ( U  e.  V  /\  W  e.  V ) )
21 usgraedgrnv 21389 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  { U ,  A }  e.  ran  E )  ->  ( U  e.  V  /\  A  e.  V ) )
2221adantrr 698 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  ->  ( U  e.  V  /\  A  e.  V ) )
23 frgranbnb.x . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  X  e.  V )
24 ax-1 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( A  =  X  ->  ( V FriendGrph  E  ->  A  =  X ) )
2524a1d 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( A  =  X  ->  (
( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
)  ->  ( V FriendGrph  E  ->  A  =  X ) ) )
2625a1d 23 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( A  =  X  ->  (
( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  ( (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) )  -> 
( V FriendGrph  E  ->  A  =  X ) ) ) )
2726a1d 23 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( A  =  X  ->  (
( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
)  ->  ( V FriendGrph  E  ->  A  =  X ) ) ) ) )
2827a1d 23 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( A  =  X  ->  ( U  =/=  W  ->  (
( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
)  ->  ( V FriendGrph  E  ->  A  =  X ) ) ) ) ) )
29 simpl 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  V USGrph  E )
3029adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  V USGrph  E )
31 simprrr 742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  W  e.  V
)
3231adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  W  e.  V )
33 simpl 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( ( U  e.  V  /\  W  e.  V )  ->  U  e.  V )
3433adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( ( ( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) )  ->  U  e.  V )
3534adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  U  e.  V
)
3635adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  U  e.  V )
37 necom 2679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( U  =/=  W  <->  W  =/=  U )
3837biimpi 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( U  =/=  W  ->  W  =/=  U )
3938adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( A  =/=  X  /\  U  =/=  W )  ->  W  =/=  U )
4039adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  W  =/=  U )
4132, 36, 403jca 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  -> 
( W  e.  V  /\  U  e.  V  /\  W  =/=  U
) )
42 simpl 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( ( X  e.  V  /\  A  e.  V )  ->  X  e.  V )
4342adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( ( ( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) )  ->  X  e.  V )
4443adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  X  e.  V
)
4544adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  X  e.  V )
46 simprlr 740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  A  e.  V
)
4746adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  A  e.  V )
48 necom 2679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( A  =/=  X  <->  X  =/=  A )
4948biimpi 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( A  =/=  X  ->  X  =/=  A )
5049adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( A  =/=  X  /\  U  =/=  W )  ->  X  =/=  A )
5150adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  X  =/=  A )
5245, 47, 513jca 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  -> 
( X  e.  V  /\  A  e.  V  /\  X  =/=  A
) )
5330, 41, 523jca 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  -> 
( V USGrph  E  /\  ( W  e.  V  /\  U  e.  V  /\  W  =/=  U
)  /\  ( X  e.  V  /\  A  e.  V  /\  X  =/= 
A ) ) )
5453ex 424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  ( ( A  =/=  X  /\  U  =/=  W )  ->  ( V USGrph  E  /\  ( W  e.  V  /\  U  e.  V  /\  W  =/= 
U )  /\  ( X  e.  V  /\  A  e.  V  /\  X  =/=  A ) ) ) )
5554adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( A  =/=  X  /\  U  =/=  W )  ->  ( V USGrph  E  /\  ( W  e.  V  /\  U  e.  V  /\  W  =/= 
U )  /\  ( X  e.  V  /\  A  e.  V  /\  X  =/=  A ) ) ) )
5655adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  ->  (
( A  =/=  X  /\  U  =/=  W
)  ->  ( V USGrph  E  /\  ( W  e.  V  /\  U  e.  V  /\  W  =/= 
U )  /\  ( X  e.  V  /\  A  e.  V  /\  X  =/=  A ) ) ) )
5756imp 419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  -> 
( V USGrph  E  /\  ( W  e.  V  /\  U  e.  V  /\  W  =/=  U
)  /\  ( X  e.  V  /\  A  e.  V  /\  X  =/= 
A ) ) )
58 prcom 3874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  { U ,  X }  =  { X ,  U }
5958eleq1i 2498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( { U ,  X }  e.  ran  E  <->  { X ,  U }  e.  ran  E )
6059biimpi 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( { U ,  X }  e.  ran  E  ->  { X ,  U }  e.  ran  E )
6160anim1i 552 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38  |-  ( ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  -> 
( { X ,  U }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )
6261ancomd 439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  -> 
( { W ,  X }  e.  ran  E  /\  { X ,  U }  e.  ran  E ) )
6362adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( { W ,  X }  e.  ran  E  /\  { X ,  U }  e.  ran  E ) )
64 prcom 3874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  { W ,  A }  =  { A ,  W }
6564eleq1i 2498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38  |-  ( { W ,  A }  e.  ran  E  <->  { A ,  W }  e.  ran  E )
6665biimpi 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( { W ,  A }  e.  ran  E  ->  { A ,  W }  e.  ran  E )
6766anim2i 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( { U ,  A }  e.  ran  E  /\  { A ,  W }  e.  ran  E ) )
6863, 67anim12i 550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  ->  (
( { W ,  X }  e.  ran  E  /\  { X ,  U }  e.  ran  E )  /\  ( { U ,  A }  e.  ran  E  /\  { A ,  W }  e.  ran  E ) ) )
6968adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  -> 
( ( { W ,  X }  e.  ran  E  /\  { X ,  U }  e.  ran  E )  /\  ( { U ,  A }  e.  ran  E  /\  { A ,  W }  e.  ran  E ) ) )
70 4cyclusnfrgra 28346 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( V USGrph  E  /\  ( W  e.  V  /\  U  e.  V  /\  W  =/=  U )  /\  ( X  e.  V  /\  A  e.  V  /\  X  =/=  A
) )  ->  (
( ( { W ,  X }  e.  ran  E  /\  { X ,  U }  e.  ran  E )  /\  ( { U ,  A }  e.  ran  E  /\  { A ,  W }  e.  ran  E ) )  ->  -.  V FriendGrph  E ) )
7157, 69, 70sylc 58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  -.  V FriendGrph  E )
7271pm2.21d 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  -> 
( V FriendGrph  E  ->  A  =  X ) )
7372ex 424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  ->  (
( A  =/=  X  /\  U  =/=  W
)  ->  ( V FriendGrph  E  ->  A  =  X ) ) )
7473com23 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  ->  ( V FriendGrph  E  ->  ( ( A  =/=  X  /\  U  =/=  W )  ->  A  =  X ) ) )
7574exp41 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( V USGrph  E  ->  ( ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V
) )  ->  (
( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( V FriendGrph  E  ->  (
( A  =/=  X  /\  U  =/=  W
)  ->  A  =  X ) ) ) ) ) )
7675com25 87 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( V USGrph  E  ->  ( V FriendGrph  E  -> 
( ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
)  ->  ( ( A  =/=  X  /\  U  =/=  W )  ->  A  =  X ) ) ) ) ) )
772, 76mpcom 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( V FriendGrph  E  ->  ( ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  -> 
( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  ( (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) )  -> 
( ( A  =/= 
X  /\  U  =/=  W )  ->  A  =  X ) ) ) ) )
7877com15 89 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( A  =/=  X  /\  U  =/=  W )  -> 
( ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
)  ->  ( V FriendGrph  E  ->  A  =  X ) ) ) ) )
7978ex 424 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( A  =/=  X  ->  ( U  =/=  W  ->  (
( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
)  ->  ( V FriendGrph  E  ->  A  =  X ) ) ) ) ) )
8028, 79pm2.61ine 2674 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( U  =/=  W  ->  (
( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
)  ->  ( V FriendGrph  E  ->  A  =  X ) ) ) ) )
8180imp 419 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
)  ->  ( V FriendGrph  E  ->  A  =  X ) ) ) )
8281com13 76 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) )  -> 
( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( V FriendGrph  E  ->  A  =  X )
) ) )
8382ex 424 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( X  e.  V  /\  A  e.  V )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  (
( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( V FriendGrph  E  ->  A  =  X )
) ) ) )
8483com25 87 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( X  e.  V  /\  A  e.  V )  ->  ( V FriendGrph  E  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  A  =  X ) ) ) ) )
8584ex 424 . . . . . . . . . . . . . . . . . . 19  |-  ( X  e.  V  ->  ( A  e.  V  ->  ( V FriendGrph  E  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ( U  =/= 
W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  A  =  X ) ) ) ) ) )
8685com23 74 . . . . . . . . . . . . . . . . . 18  |-  ( X  e.  V  ->  ( V FriendGrph  E  ->  ( A  e.  V  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ( U  =/= 
W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  A  =  X ) ) ) ) ) )
8723, 1, 86sylc 58 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( A  e.  V  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  A  =  X ) ) ) ) )
8887com13 76 . . . . . . . . . . . . . . . 16  |-  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( A  e.  V  ->  ( ph  ->  (
( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  (
( U  e.  V  /\  W  e.  V
)  ->  A  =  X ) ) ) ) )
8988adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  ->  ( A  e.  V  ->  ( ph  ->  ( ( U  =/= 
W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  A  =  X ) ) ) ) )
9089com12 29 . . . . . . . . . . . . . 14  |-  ( A  e.  V  ->  (
( V USGrph  E  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  ->  ( ph  ->  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  (
( U  e.  V  /\  W  e.  V
)  ->  A  =  X ) ) ) ) )
9190adantl 453 . . . . . . . . . . . . 13  |-  ( ( U  e.  V  /\  A  e.  V )  ->  ( ( V USGrph  E  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  ->  ( ph  ->  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  A  =  X ) ) ) ) )
9222, 91mpcom 34 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E ) )  ->  ( ph  ->  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  (
( U  e.  V  /\  W  e.  V
)  ->  A  =  X ) ) ) )
9392ex 424 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( ph  ->  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  A  =  X ) ) ) ) )
9493com25 87 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( ( U  e.  V  /\  W  e.  V )  ->  ( ph  ->  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X )
) ) ) )
9594com14 84 . . . . . . . . 9  |-  ( ( U  =/=  W  /\  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  ( ph  ->  ( V USGrph  E  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X ) ) ) ) )
9695ex 424 . . . . . . . 8  |-  ( U  =/=  W  ->  (
( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  ->  ( ( U  e.  V  /\  W  e.  V )  ->  ( ph  ->  ( V USGrph  E  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X )
) ) ) ) )
9796com15 89 . . . . . . 7  |-  ( V USGrph  E  ->  ( ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  -> 
( ( U  e.  V  /\  W  e.  V )  ->  ( ph  ->  ( U  =/= 
W  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X )
) ) ) ) )
9897adantr 452 . . . . . 6  |-  ( ( V USGrph  E  /\  ( U  e.  D  /\  W  e.  D )
)  ->  ( ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  -> 
( ( U  e.  V  /\  W  e.  V )  ->  ( ph  ->  ( U  =/= 
W  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X )
) ) ) ) )
9914, 20, 98mp2d 43 . . . . 5  |-  ( ( V USGrph  E  /\  ( U  e.  D  /\  W  e.  D )
)  ->  ( ph  ->  ( U  =/=  W  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X ) ) ) )
10099ex 424 . . . 4  |-  ( V USGrph  E  ->  ( ( U  e.  D  /\  W  e.  D )  ->  ( ph  ->  ( U  =/= 
W  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X )
) ) ) )
101100com23 74 . . 3  |-  ( V USGrph  E  ->  ( ph  ->  ( ( U  e.  D  /\  W  e.  D
)  ->  ( U  =/=  W  ->  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X )
) ) ) )
1023, 101mpcom 34 . 2  |-  ( ph  ->  ( ( U  e.  D  /\  W  e.  D )  ->  ( U  =/=  W  ->  (
( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X ) ) ) )
1031023imp 1147 1  |-  ( (
ph  /\  ( U  e.  D  /\  W  e.  D )  /\  U  =/=  W )  ->  (
( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   {cpr 3807   <.cop 3809   class class class wbr 4204   ran crn 4871  (class class class)co 6073   USGrph cusg 21357   Neighbors cnbgra 21422   FriendGrph cfrgra 28315
This theorem is referenced by:  frgrancvvdeqlemB  28364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611  df-usgra 21359  df-nbgra 21425  df-frgra 28316
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