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Theorem frgraun 25413
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 25412 . 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 4633 . . . . 5  |-  { A ,  b }  e.  _V
3 prex 4633 . . . . 5  |-  { b ,  C }  e.  _V
42, 3prss 4126 . . . 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 2994 . 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 367    /\ w3a 974    e. wcel 1842    =/= wne 2598   E!wreu 2756    C_ wss 3414   {cpr 3974   class class class wbr 4395   ran crn 4824   FriendGrph cfrgra 25405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-dm 4833  df-rn 4834  df-frgra 25406
This theorem is referenced by:  frgrancvvdeqlemC  25456  frgraeu  25471  frg2woteu  25472  numclwwlk2lem1  25519
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