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Theorem frgraun 31714
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 31713 . 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 4682 . . . . 5  |-  { A ,  b }  e.  _V
3 prex 4682 . . . . 5  |-  { b ,  C }  e.  _V
42, 3prss 4174 . . . 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 3041 . 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 968    e. wcel 1762    =/= wne 2655   E!wreu 2809    C_ wss 3469   {cpr 4022   class class class wbr 4440   ran crn 4993   FriendGrph cfrgra 31706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003  df-frgra 31707
This theorem is referenced by:  frgrancvvdeqlemC  31758  frgraeu  31773  frg2woteu  31774  numclwwlk2lem1  31821
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