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Theorem frgraunss 28099
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
frgraunss  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E ) )
Distinct variable groups:    A, b    C, b    E, b    V, b

Proof of Theorem frgraunss
Dummy variables  a 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgrapr 28095 . 2  |-  ( V FriendGrph  E  ->  ( V USGrph  E  /\  A. a  e.  V  A. c  e.  ( V  \  { a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_  ran  E ) )
2 sneq 3785 . . . . . . . . . . 11  |-  ( a  =  A  ->  { a }  =  { A } )
32difeq2d 3425 . . . . . . . . . 10  |-  ( a  =  A  ->  ( V  \  { a } )  =  ( V 
\  { A }
) )
4 preq2 3844 . . . . . . . . . . . . 13  |-  ( a  =  A  ->  { b ,  a }  =  { b ,  A } )
54preq1d 3849 . . . . . . . . . . . 12  |-  ( a  =  A  ->  { {
b ,  a } ,  { b ,  c } }  =  { { b ,  A } ,  { b ,  c } }
)
65sseq1d 3335 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  <->  { { b ,  A } ,  {
b ,  c } }  C_  ran  E ) )
76reubidv 2852 . . . . . . . . . 10  |-  ( a  =  A  ->  ( E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  <->  E! b  e.  V  { { b ,  A } ,  { b ,  c } }  C_ 
ran  E ) )
83, 7raleqbidv 2876 . . . . . . . . 9  |-  ( a  =  A  ->  ( A. c  e.  ( V  \  { a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_  ran  E  <->  A. c  e.  ( V  \  { A }
) E! b  e.  V  { { b ,  A } ,  { b ,  c } }  C_  ran  E ) )
98rspcva 3010 . . . . . . . 8  |-  ( ( A  e.  V  /\  A. a  e.  V  A. c  e.  ( V  \  { a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E )  ->  A. c  e.  ( V  \  { A }
) E! b  e.  V  { { b ,  A } ,  { b ,  c } }  C_  ran  E )
10 elsni 3798 . . . . . . . . . . . . . . 15  |-  ( C  e.  { A }  ->  C  =  A )
1110eqcomd 2409 . . . . . . . . . . . . . 14  |-  ( C  e.  { A }  ->  A  =  C )
1211necon3ai 2607 . . . . . . . . . . . . 13  |-  ( A  =/=  C  ->  -.  C  e.  { A } )
1312anim2i 553 . . . . . . . . . . . 12  |-  ( ( C  e.  V  /\  A  =/=  C )  -> 
( C  e.  V  /\  -.  C  e.  { A } ) )
14 eldif 3290 . . . . . . . . . . . 12  |-  ( C  e.  ( V  \  { A } )  <->  ( C  e.  V  /\  -.  C  e.  { A } ) )
1513, 14sylibr 204 . . . . . . . . . . 11  |-  ( ( C  e.  V  /\  A  =/=  C )  ->  C  e.  ( V  \  { A } ) )
16 preq2 3844 . . . . . . . . . . . . . . . . 17  |-  ( c  =  C  ->  { b ,  c }  =  { b ,  C } )
1716preq2d 3850 . . . . . . . . . . . . . . . 16  |-  ( c  =  C  ->  { {
b ,  A } ,  { b ,  c } }  =  { { b ,  A } ,  { b ,  C } } )
1817sseq1d 3335 . . . . . . . . . . . . . . 15  |-  ( c  =  C  ->  ( { { b ,  A } ,  { b ,  c } }  C_ 
ran  E  <->  { { b ,  A } ,  {
b ,  C } }  C_  ran  E ) )
1918reubidv 2852 . . . . . . . . . . . . . 14  |-  ( c  =  C  ->  ( E! b  e.  V  { { b ,  A } ,  { b ,  c } }  C_ 
ran  E  <->  E! b  e.  V  { { b ,  A } ,  { b ,  C } }  C_  ran  E ) )
2019rspcva 3010 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( V 
\  { A }
)  /\  A. c  e.  ( V  \  { A } ) E! b  e.  V  { {
b ,  A } ,  { b ,  c } }  C_  ran  E )  ->  E! b  e.  V  { { b ,  A } ,  { b ,  C } }  C_  ran  E
)
21 prcom 3842 . . . . . . . . . . . . . . . 16  |-  { b ,  A }  =  { A ,  b }
2221preq1i 3846 . . . . . . . . . . . . . . 15  |-  { {
b ,  A } ,  { b ,  C } }  =  { { A ,  b } ,  { b ,  C } }
2322sseq1i 3332 . . . . . . . . . . . . . 14  |-  ( { { b ,  A } ,  { b ,  C } }  C_  ran  E  <->  { { A , 
b } ,  {
b ,  C } }  C_  ran  E )
2423reubii 2854 . . . . . . . . . . . . 13  |-  ( E! b  e.  V  { { b ,  A } ,  { b ,  C } }  C_  ran  E  <->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_ 
ran  E )
2520, 24sylib 189 . . . . . . . . . . . 12  |-  ( ( C  e.  ( V 
\  { A }
)  /\  A. c  e.  ( V  \  { A } ) E! b  e.  V  { {
b ,  A } ,  { b ,  c } }  C_  ran  E )  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E
)
2625ex 424 . . . . . . . . . . 11  |-  ( C  e.  ( V  \  { A } )  -> 
( A. c  e.  ( V  \  { A } ) E! b  e.  V  { {
b ,  A } ,  { b ,  c } }  C_  ran  E  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E
) )
2715, 26syl 16 . . . . . . . . . 10  |-  ( ( C  e.  V  /\  A  =/=  C )  -> 
( A. c  e.  ( V  \  { A } ) E! b  e.  V  { {
b ,  A } ,  { b ,  c } }  C_  ran  E  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E
) )
2827ex 424 . . . . . . . . 9  |-  ( C  e.  V  ->  ( A  =/=  C  ->  ( A. c  e.  ( V  \  { A }
) E! b  e.  V  { { b ,  A } ,  { b ,  c } }  C_  ran  E  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E
) ) )
2928com13 76 . . . . . . . 8  |-  ( A. c  e.  ( V  \  { A } ) E! b  e.  V  { { b ,  A } ,  { b ,  c } }  C_ 
ran  E  ->  ( A  =/=  C  ->  ( C  e.  V  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E ) ) )
309, 29syl 16 . . . . . . 7  |-  ( ( A  e.  V  /\  A. a  e.  V  A. c  e.  ( V  \  { a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E )  -> 
( A  =/=  C  ->  ( C  e.  V  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_ 
ran  E ) ) )
3130ex 424 . . . . . 6  |-  ( A  e.  V  ->  ( A. a  e.  V  A. c  e.  ( V  \  { a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_  ran  E  ->  ( A  =/= 
C  ->  ( C  e.  V  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E
) ) ) )
3231com24 83 . . . . 5  |-  ( A  e.  V  ->  ( C  e.  V  ->  ( A  =/=  C  -> 
( A. a  e.  V  A. c  e.  ( V  \  {
a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E
) ) ) )
33323imp 1147 . . . 4  |-  ( ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  -> 
( A. a  e.  V  A. c  e.  ( V  \  {
a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E
) )
3433com12 29 . . 3  |-  ( A. a  e.  V  A. c  e.  ( V  \  { a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_ 
ran  E ) )
3534adantl 453 . 2  |-  ( ( V USGrph  E  /\  A. a  e.  V  A. c  e.  ( V  \  {
a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E )  -> 
( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E ) )
361, 35syl 16 1  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran  E ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E!wreu 2668    \ cdif 3277    C_ wss 3280   {csn 3774   {cpr 3775   class class class wbr 4172   ran crn 4838   USGrph cusg 21318   FriendGrph cfrgra 28092
This theorem is referenced by:  frgraun  28100  4cyclusnfrgra  28123  frgrancvvdeqlem3  28135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-frgra 28093
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