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Theorem frgraunss 30590
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 30586 . 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 3890 . . . . . . . . . . 11  |-  ( a  =  A  ->  { a }  =  { A } )
32difeq2d 3477 . . . . . . . . . 10  |-  ( a  =  A  ->  ( V  \  { a } )  =  ( V 
\  { A }
) )
4 preq2 3958 . . . . . . . . . . . . 13  |-  ( a  =  A  ->  { b ,  a }  =  { b ,  A } )
54preq1d 3963 . . . . . . . . . . . 12  |-  ( a  =  A  ->  { {
b ,  a } ,  { b ,  c } }  =  { { b ,  A } ,  { b ,  c } }
)
65sseq1d 3386 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  <->  { { b ,  A } ,  {
b ,  c } }  C_  ran  E ) )
76reubidv 2908 . . . . . . . . . 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 2934 . . . . . . . . 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 3074 . . . . . . . 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 3905 . . . . . . . . . . . . . . 15  |-  ( C  e.  { A }  ->  C  =  A )
1110eqcomd 2448 . . . . . . . . . . . . . 14  |-  ( C  e.  { A }  ->  A  =  C )
1211necon3ai 2654 . . . . . . . . . . . . 13  |-  ( A  =/=  C  ->  -.  C  e.  { A } )
1312anim2i 569 . . . . . . . . . . . 12  |-  ( ( C  e.  V  /\  A  =/=  C )  -> 
( C  e.  V  /\  -.  C  e.  { A } ) )
14 eldif 3341 . . . . . . . . . . . 12  |-  ( C  e.  ( V  \  { A } )  <->  ( C  e.  V  /\  -.  C  e.  { A } ) )
1513, 14sylibr 212 . . . . . . . . . . 11  |-  ( ( C  e.  V  /\  A  =/=  C )  ->  C  e.  ( V  \  { A } ) )
16 preq2 3958 . . . . . . . . . . . . . . . . 17  |-  ( c  =  C  ->  { b ,  c }  =  { b ,  C } )
1716preq2d 3964 . . . . . . . . . . . . . . . 16  |-  ( c  =  C  ->  { {
b ,  A } ,  { b ,  c } }  =  { { b ,  A } ,  { b ,  C } } )
1817sseq1d 3386 . . . . . . . . . . . . . . 15  |-  ( c  =  C  ->  ( { { b ,  A } ,  { b ,  c } }  C_ 
ran  E  <->  { { b ,  A } ,  {
b ,  C } }  C_  ran  E ) )
1918reubidv 2908 . . . . . . . . . . . . . 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 3074 . . . . . . . . . . . . 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 3956 . . . . . . . . . . . . . . . 16  |-  { b ,  A }  =  { A ,  b }
2221preq1i 3960 . . . . . . . . . . . . . . 15  |-  { {
b ,  A } ,  { b ,  C } }  =  { { A ,  b } ,  { b ,  C } }
2322sseq1i 3383 . . . . . . . . . . . . . 14  |-  ( { { b ,  A } ,  { b ,  C } }  C_  ran  E  <->  { { A , 
b } ,  {
b ,  C } }  C_  ran  E )
2423reubii 2910 . . . . . . . . . . . . 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 196 . . . . . . . . . . . 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 434 . . . . . . . . . . 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 434 . . . . . . . . 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 80 . . . . . . . 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 434 . . . . . 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 87 . . . . 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 1181 . . . 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 31 . . 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 466 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2609   A.wral 2718   E!wreu 2720    \ cdif 3328    C_ wss 3331   {csn 3880   {cpr 3882   class class class wbr 4295   ran crn 4844   USGrph cusg 23267   FriendGrph cfrgra 30583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pr 4534
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-br 4296  df-opab 4354  df-xp 4849  df-rel 4850  df-cnv 4851  df-dm 4853  df-rn 4854  df-frgra 30584
This theorem is referenced by:  frgraun  30591  4cyclusnfrgra  30614  frgrancvvdeqlem3  30628
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