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Theorem frgraunss 25121
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 25117 . 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 4042 . . . . . . . . . . 11  |-  ( a  =  A  ->  { a }  =  { A } )
32difeq2d 3618 . . . . . . . . . 10  |-  ( a  =  A  ->  ( V  \  { a } )  =  ( V 
\  { A }
) )
4 preq2 4112 . . . . . . . . . . . . 13  |-  ( a  =  A  ->  { b ,  a }  =  { b ,  A } )
54preq1d 4117 . . . . . . . . . . . 12  |-  ( a  =  A  ->  { {
b ,  a } ,  { b ,  c } }  =  { { b ,  A } ,  { b ,  c } }
)
65sseq1d 3526 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  <->  { { b ,  A } ,  {
b ,  c } }  C_  ran  E ) )
76reubidv 3042 . . . . . . . . . 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 3068 . . . . . . . . 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 3208 . . . . . . . 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 4057 . . . . . . . . . . . . . . 15  |-  ( C  e.  { A }  ->  C  =  A )
1110eqcomd 2465 . . . . . . . . . . . . . 14  |-  ( C  e.  { A }  ->  A  =  C )
1211necon3ai 2685 . . . . . . . . . . . . 13  |-  ( A  =/=  C  ->  -.  C  e.  { A } )
1312anim2i 569 . . . . . . . . . . . 12  |-  ( ( C  e.  V  /\  A  =/=  C )  -> 
( C  e.  V  /\  -.  C  e.  { A } ) )
14 eldif 3481 . . . . . . . . . . . 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 4112 . . . . . . . . . . . . . . . . 17  |-  ( c  =  C  ->  { b ,  c }  =  { b ,  C } )
1716preq2d 4118 . . . . . . . . . . . . . . . 16  |-  ( c  =  C  ->  { {
b ,  A } ,  { b ,  c } }  =  { { b ,  A } ,  { b ,  C } } )
1817sseq1d 3526 . . . . . . . . . . . . . . 15  |-  ( c  =  C  ->  ( { { b ,  A } ,  { b ,  c } }  C_ 
ran  E  <->  { { b ,  A } ,  {
b ,  C } }  C_  ran  E ) )
1918reubidv 3042 . . . . . . . . . . . . . 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 3208 . . . . . . . . . . . . 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 4110 . . . . . . . . . . . . . . . 16  |-  { b ,  A }  =  { A ,  b }
2221preq1i 4114 . . . . . . . . . . . . . . 15  |-  { {
b ,  A } ,  { b ,  C } }  =  { { A ,  b } ,  { b ,  C } }
2322sseq1i 3523 . . . . . . . . . . . . . 14  |-  ( { { b ,  A } ,  { b ,  C } }  C_  ran  E  <->  { { A , 
b } ,  {
b ,  C } }  C_  ran  E )
2423reubii 3044 . . . . . . . . . . . . 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 1190 . . . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E!wreu 2809    \ cdif 3468    C_ wss 3471   {csn 4032   {cpr 4034   class class class wbr 4456   ran crn 5009   USGrph cusg 24456   FriendGrph cfrgra 25114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-frgra 25115
This theorem is referenced by:  frgraun  25122  4cyclusnfrgra  25145  frgrancvvdeqlem3  25158
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