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Theorem frgraunss 25727
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 25723 . 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 4014 . . . . . . . . . . 11  |-  ( a  =  A  ->  { a }  =  { A } )
32difeq2d 3589 . . . . . . . . . 10  |-  ( a  =  A  ->  ( V  \  { a } )  =  ( V 
\  { A }
) )
4 preq2 4086 . . . . . . . . . . . . 13  |-  ( a  =  A  ->  { b ,  a }  =  { b ,  A } )
54preq1d 4091 . . . . . . . . . . . 12  |-  ( a  =  A  ->  { {
b ,  a } ,  { b ,  c } }  =  { { b ,  A } ,  { b ,  c } }
)
65sseq1d 3497 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  <->  { { b ,  A } ,  {
b ,  c } }  C_  ran  E ) )
76reubidv 3015 . . . . . . . . . 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 3041 . . . . . . . . 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 3186 . . . . . . . 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 4029 . . . . . . . . . . . . . . 15  |-  ( C  e.  { A }  ->  C  =  A )
1110eqcomd 2431 . . . . . . . . . . . . . 14  |-  ( C  e.  { A }  ->  A  =  C )
1211necon3ai 2653 . . . . . . . . . . . . 13  |-  ( A  =/=  C  ->  -.  C  e.  { A } )
1312anim2i 572 . . . . . . . . . . . 12  |-  ( ( C  e.  V  /\  A  =/=  C )  -> 
( C  e.  V  /\  -.  C  e.  { A } ) )
14 eldif 3452 . . . . . . . . . . . 12  |-  ( C  e.  ( V  \  { A } )  <->  ( C  e.  V  /\  -.  C  e.  { A } ) )
1513, 14sylibr 216 . . . . . . . . . . 11  |-  ( ( C  e.  V  /\  A  =/=  C )  ->  C  e.  ( V  \  { A } ) )
16 preq2 4086 . . . . . . . . . . . . . . . . 17  |-  ( c  =  C  ->  { b ,  c }  =  { b ,  C } )
1716preq2d 4092 . . . . . . . . . . . . . . . 16  |-  ( c  =  C  ->  { {
b ,  A } ,  { b ,  c } }  =  { { b ,  A } ,  { b ,  C } } )
1817sseq1d 3497 . . . . . . . . . . . . . . 15  |-  ( c  =  C  ->  ( { { b ,  A } ,  { b ,  c } }  C_ 
ran  E  <->  { { b ,  A } ,  {
b ,  C } }  C_  ran  E ) )
1918reubidv 3015 . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . 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 4084 . . . . . . . . . . . . . . . 16  |-  { b ,  A }  =  { A ,  b }
2221preq1i 4088 . . . . . . . . . . . . . . 15  |-  { {
b ,  A } ,  { b ,  C } }  =  { { A ,  b } ,  { b ,  C } }
2322sseq1i 3494 . . . . . . . . . . . . . 14  |-  ( { { b ,  A } ,  { b ,  C } }  C_  ran  E  <->  { { A , 
b } ,  {
b ,  C } }  C_  ran  E )
2423reubii 3017 . . . . . . . . . . . . 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 200 . . . . . . . . . . . 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 436 . . . . . . . . . . 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 17 . . . . . . . . . 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 436 . . . . . . . . 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 84 . . . . . . . 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 17 . . . . . . 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 436 . . . . . 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 91 . . . . 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 1200 . . . 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 33 . . 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 468 . 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 17 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1873    =/= wne 2619   A.wral 2776   E!wreu 2778    \ cdif 3439    C_ wss 3442   {csn 4004   {cpr 4006   class class class wbr 4429   ran crn 4860   USGrph cusg 25061   FriendGrph cfrgra 25720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-sep 4552  ax-nul 4561  ax-pr 4666
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3087  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3918  df-sn 4005  df-pr 4007  df-op 4011  df-br 4430  df-opab 4489  df-xp 4865  df-rel 4866  df-cnv 4867  df-dm 4869  df-rn 4870  df-frgra 25721
This theorem is referenced by:  frgraun  25728  4cyclusnfrgra  25751  frgrancvvdeqlem3  25764
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