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Theorem frgranbnb 25601
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 25573 . . . 4  |-  ( V FriendGrph  E  ->  V USGrph  E )
31, 2syl 17 . . 3  |-  ( ph  ->  V USGrph  E )
4 frgranbnb.nx . . . . . . . . . 10  |-  D  =  ( <. V ,  E >. Neighbors  X )
54eleq2i 2507 . . . . . . . . 9  |-  ( U  e.  D  <->  U  e.  ( <. V ,  E >. Neighbors  X ) )
6 nbgraeledg 25011 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( U  e.  ( <. V ,  E >. Neighbors  X )  <->  { U ,  X }  e.  ran  E ) )
76biimpd 210 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( U  e.  ( <. V ,  E >. Neighbors  X )  ->  { U ,  X }  e.  ran  E ) )
85, 7syl5bi 220 . . . . . . . 8  |-  ( V USGrph  E  ->  ( U  e.  D  ->  { U ,  X }  e.  ran  E ) )
94eleq2i 2507 . . . . . . . . 9  |-  ( W  e.  D  <->  W  e.  ( <. V ,  E >. Neighbors  X ) )
10 nbgraeledg 25011 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( W  e.  ( <. V ,  E >. Neighbors  X )  <->  { W ,  X }  e.  ran  E ) )
1110biimpd 210 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( W  e.  ( <. V ,  E >. Neighbors  X )  ->  { W ,  X }  e.  ran  E ) )
129, 11syl5bi 220 . . . . . . . 8  |-  ( V USGrph  E  ->  ( W  e.  D  ->  { W ,  X }  e.  ran  E ) )
138, 12anim12d 565 . . . . . . 7  |-  ( V USGrph  E  ->  ( ( U  e.  D  /\  W  e.  D )  ->  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) ) )
1413imp 430 . . . . . 6  |-  ( ( V USGrph  E  /\  ( U  e.  D  /\  W  e.  D )
)  ->  ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )
15 nbgraisvtx 25012 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( U  e.  ( <. V ,  E >. Neighbors  X )  ->  U  e.  V ) )
165, 15syl5bi 220 . . . . . . . 8  |-  ( V USGrph  E  ->  ( U  e.  D  ->  U  e.  V ) )
17 nbgraisvtx 25012 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( W  e.  ( <. V ,  E >. Neighbors  X )  ->  W  e.  V ) )
189, 17syl5bi 220 . . . . . . . 8  |-  ( V USGrph  E  ->  ( W  e.  D  ->  W  e.  V ) )
1916, 18anim12d 565 . . . . . . 7  |-  ( V USGrph  E  ->  ( ( U  e.  D  /\  W  e.  D )  ->  ( U  e.  V  /\  W  e.  V )
) )
2019imp 430 . . . . . 6  |-  ( ( V USGrph  E  /\  ( U  e.  D  /\  W  e.  D )
)  ->  ( U  e.  V  /\  W  e.  V ) )
21 usgraedgrnv 24958 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  { U ,  A }  e.  ran  E )  ->  ( U  e.  V  /\  A  e.  V ) )
2221adantrr 721 . . . . . . . . . . . . 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 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( A  =  X  ->  ( V FriendGrph  E  ->  A  =  X ) )
25242a1d 27 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 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 ) ) ) )
26252a1d 27 . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
27 simpl 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  V USGrph  E )
2827adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  V USGrph  E )
29 simprrr 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  W  e.  V
)
3029adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  W  e.  V )
31 simpl 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( ( U  e.  V  /\  W  e.  V )  ->  U  e.  V )
3231adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( ( ( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) )  ->  U  e.  V )
3332adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  U  e.  V
)
3433adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  U  e.  V )
35 necom 2700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( U  =/=  W  <->  W  =/=  U )
3635biimpi 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( U  =/=  W  ->  W  =/=  U )
3736adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( A  =/=  X  /\  U  =/=  W )  ->  W  =/=  U )
3837adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  W  =/=  U )
3930, 34, 383jca 1185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
) )
40 simpl 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( ( X  e.  V  /\  A  e.  V )  ->  X  e.  V )
4140adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( ( ( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) )  ->  X  e.  V )
4241adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  X  e.  V
)
4342adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  X  e.  V )
44 simprlr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( V USGrph  E  /\  (
( X  e.  V  /\  A  e.  V
)  /\  ( U  e.  V  /\  W  e.  V ) ) )  ->  A  e.  V
)
4544adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  A  e.  V )
46 necom 2700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( A  =/=  X  <->  X  =/=  A )
4746biimpi 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( A  =/=  X  ->  X  =/=  A )
4847adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( A  =/=  X  /\  U  =/=  W )  ->  X  =/=  A )
4948adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( ( V USGrph  E  /\  ( ( X  e.  V  /\  A  e.  V )  /\  ( U  e.  V  /\  W  e.  V )
) )  /\  ( A  =/=  X  /\  U  =/=  W ) )  ->  X  =/=  A )
5043, 45, 493jca 1185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
) )
5128, 39, 503jca 1185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
5251ex 435 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
5352adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
5453adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
5554imp 430 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
56 prcom 4081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  { U ,  X }  =  { X ,  U }
5756eleq1i 2506 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( { U ,  X }  e.  ran  E  <->  { X ,  U }  e.  ran  E )
5857biimpi 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( { U ,  X }  e.  ran  E  ->  { X ,  U }  e.  ran  E )
5958anim1i 570 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38  |-  ( ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  -> 
( { X ,  U }  e.  ran  E  /\  { W ,  X }  e.  ran  E ) )
6059ancomd 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( ( { U ,  X }  e.  ran  E  /\  { W ,  X }  e.  ran  E )  -> 
( { W ,  X }  e.  ran  E  /\  { X ,  U }  e.  ran  E ) )
6160adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) )
62 prcom 4081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  { W ,  A }  =  { A ,  W }
6362eleq1i 2506 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38  |-  ( { W ,  A }  e.  ran  E  <->  { A ,  W }  e.  ran  E )
6463biimpi 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( { W ,  A }  e.  ran  E  ->  { A ,  W }  e.  ran  E )
6564anim2i 571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  -> 
( { U ,  A }  e.  ran  E  /\  { A ,  W }  e.  ran  E ) )
6661, 65anim12i 568 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
6766adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
68 4cyclusnfrgra 25600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) )
6955, 67, 68sylc 62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )
7069pm2.21d 109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) )
7170ex 435 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
7271com23 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
7372exp41 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
7473com25 94 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
752, 74mpcom 37 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
7675com15 96 . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
7776ex 435 . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
7826, 77pm2.61ine 2744 . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
7978imp 430 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
8079com13 83 . . . . . . . . . . . . . . . . . . . . . 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 )
) ) )
8180ex 435 . . . . . . . . . . . . . . . . . . . . 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 )
) ) ) )
8281com25 94 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
8382ex 435 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
8483com23 81 . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
8523, 1, 84sylc 62 . . . . . . . . . . . . . . . . 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 ) ) ) ) )
8685com13 83 . . . . . . . . . . . . . . . 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 ) ) ) ) )
8786adantl 467 . . . . . . . . . . . . . . 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 ) ) ) ) )
8887com12 32 . . . . . . . . . . . . . 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 ) ) ) ) )
8988adantl 467 . . . . . . . . . . . . 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 ) ) ) ) )
9022, 89mpcom 37 . . . . . . . . . . . 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 ) ) ) )
9190ex 435 . . . . . . . . . . 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 ) ) ) ) )
9291com25 94 . . . . . . . . . 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 )
) ) ) )
9392com14 91 . . . . . . . . 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 ) ) ) ) )
9493ex 435 . . . . . . . 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 )
) ) ) ) )
9594com15 96 . . . . . . 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 )
) ) ) ) )
9695adantr 466 . . . . . 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 )
) ) ) ) )
9714, 20, 96mp2d 46 . . . . 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 ) ) ) )
9897ex 435 . . . 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 )
) ) ) )
9998com23 81 . . 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 )
) ) ) )
1003, 99mpcom 37 . 2  |-  ( ph  ->  ( ( U  e.  D  /\  W  e.  D )  ->  ( U  =/=  W  ->  (
( { U ,  A }  e.  ran  E  /\  { W ,  A }  e.  ran  E )  ->  A  =  X ) ) ) )
1011003imp 1199 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   {cpr 4004   <.cop 4008   class class class wbr 4426   ran crn 4855  (class class class)co 6305   USGrph cusg 24911   Neighbors cnbgra 24998   FriendGrph cfrgra 25569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-hash 12513  df-usgra 24914  df-nbgra 25001  df-frgra 25570
This theorem is referenced by:  frgrancvvdeqlemB  25619
  Copyright terms: Public domain W3C validator