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Theorem frgraun 24868
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 24867 . 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 4679 . . . . 5  |-  { A ,  b }  e.  _V
3 prex 4679 . . . . 5  |-  { b ,  C }  e.  _V
42, 3prss 4169 . . . 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 3030 . 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 974    e. wcel 1804    =/= wne 2638   E!wreu 2795    C_ wss 3461   {cpr 4016   class class class wbr 4437   ran crn 4990   FriendGrph cfrgra 24860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-frgra 24861
This theorem is referenced by:  frgrancvvdeqlemC  24911  frgraeu  24926  frg2woteu  24927  numclwwlk2lem1  24974
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