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Theorem frgraun 30728
Description: Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Assertion
Ref Expression
frgraun  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! b  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
Distinct variable groups:    A, b    C, b    E, b    V, b

Proof of Theorem frgraun
StepHypRef Expression
1 frgraunss 30727 . 2  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E ) )
2 prex 4634 . . . . 5  |-  { A ,  b }  e.  _V
3 prex 4634 . . . . 5  |-  { b ,  C }  e.  _V
42, 3prss 4127 . . . 4  |-  ( ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E )  <->  { { A ,  b } ,  { b ,  C } }  C_ 
ran  E )
54bicomi 202 . . 3  |-  ( { { A ,  b } ,  { b ,  C } }  C_ 
ran  E  <->  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) )
65reubii 3005 . 2  |-  ( E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E  <->  E! b  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) )
71, 6syl6ib 226 1  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! b  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2644   E!wreu 2797    C_ wss 3428   {cpr 3979   class class class wbr 4392   ran crn 4941   FriendGrph cfrgra 30720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951  df-frgra 30721
This theorem is referenced by:  frgrancvvdeqlemC  30772  frgraeu  30787  frg2woteu  30788  numclwwlk2lem1  30835
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