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Theorem frgra2v 28451
Description: Any graph with two (different) vertices is not a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
Assertion
Ref Expression
frgra2v  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B )  ->  -.  { A ,  B } FriendGrph  E )

Proof of Theorem frgra2v
Dummy variables  k 
l  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . . . . . . . . . . . . . . . . 18  |-  A  =  A
2 prex 4408 . . . . . . . . . . . . . . . . . . . . 21  |-  { A ,  B }  e.  _V
3 prex 4408 . . . . . . . . . . . . . . . . . . . . 21  |-  { A ,  A }  e.  _V
42, 3prss 3954 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( { A ,  B }  e.  ran  E  /\  { A ,  A }  e.  ran  E )  <->  { { A ,  B } ,  { A ,  A } }  C_  ran  E )
5 usgraedgrn 21403 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { A ,  B } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
65neneqd 2619 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  -.  A  =  A )
76expcom 426 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B } USGrph  E  ->  -.  A  =  A ) )
87adantl 454 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( { A ,  B }  e.  ran  E  /\  { A ,  A }  e.  ran  E )  -> 
( { A ,  B } USGrph  E  ->  -.  A  =  A )
)
94, 8sylbir 206 . . . . . . . . . . . . . . . . . . 19  |-  ( { { A ,  B } ,  { A ,  A } }  C_  ran  E  ->  ( { A ,  B } USGrph  E  ->  -.  A  =  A ) )
109com12 30 . . . . . . . . . . . . . . . . . 18  |-  ( { A ,  B } USGrph  E  ->  ( { { A ,  B } ,  { A ,  A } }  C_  ran  E  ->  -.  A  =  A ) )
111, 10mt2i 113 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  B } USGrph  E  ->  -.  { { A ,  B } ,  { A ,  A } }  C_  ran  E )
12 eqid 2438 . . . . . . . . . . . . . . . . . 18  |-  B  =  B
13 prex 4408 . . . . . . . . . . . . . . . . . . . . 21  |-  { B ,  B }  e.  _V
14 prex 4408 . . . . . . . . . . . . . . . . . . . . 21  |-  { B ,  A }  e.  _V
1513, 14prss 3954 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( { B ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E )  <->  { { B ,  B } ,  { B ,  A } }  C_  ran  E )
16 usgraedgrn 21403 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { A ,  B } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  B  =/=  B )
1716neneqd 2619 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  -.  B  =  B )
1817expcom 426 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { B ,  B }  e.  ran  E  ->  ( { A ,  B } USGrph  E  ->  -.  B  =  B ) )
1918adantr 453 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( { B ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E )  -> 
( { A ,  B } USGrph  E  ->  -.  B  =  B )
)
2015, 19sylbir 206 . . . . . . . . . . . . . . . . . . 19  |-  ( { { B ,  B } ,  { B ,  A } }  C_  ran  E  ->  ( { A ,  B } USGrph  E  ->  -.  B  =  B ) )
2120com12 30 . . . . . . . . . . . . . . . . . 18  |-  ( { A ,  B } USGrph  E  ->  ( { { B ,  B } ,  { B ,  A } }  C_  ran  E  ->  -.  B  =  B ) )
2212, 21mt2i 113 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  B } USGrph  E  ->  -.  { { B ,  B } ,  { B ,  A } }  C_  ran  E )
2311, 22jca 520 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B } USGrph  E  ->  ( -.  { { A ,  B } ,  { A ,  A } }  C_  ran  E  /\  -.  { { B ,  B } ,  { B ,  A } }  C_  ran  E ) )
2423adantl 454 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( -.  { { A ,  B } ,  { A ,  A } }  C_  ran  E  /\  -.  { { B ,  B } ,  { B ,  A } }  C_  ran  E
) )
25 preq1 3885 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
26 preq1 3885 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  A  ->  { x ,  A }  =  { A ,  A }
)
2725, 26preq12d 3893 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  A  ->  { {
x ,  B } ,  { x ,  A } }  =  { { A ,  B } ,  { A ,  A } } )
2827sseq1d 3377 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  A  ->  ( { { x ,  B } ,  { x ,  A } }  C_  ran  E  <->  { { A ,  B } ,  { A ,  A } }  C_  ran  E ) )
2928notbid 287 . . . . . . . . . . . . . . . . 17  |-  ( x  =  A  ->  ( -.  { { x ,  B } ,  {
x ,  A } }  C_  ran  E  <->  -.  { { A ,  B } ,  { A ,  A } }  C_  ran  E
) )
30 preq1 3885 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  B  ->  { x ,  B }  =  { B ,  B }
)
31 preq1 3885 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  B  ->  { x ,  A }  =  { B ,  A }
)
3230, 31preq12d 3893 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  B  ->  { {
x ,  B } ,  { x ,  A } }  =  { { B ,  B } ,  { B ,  A } } )
3332sseq1d 3377 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  B  ->  ( { { x ,  B } ,  { x ,  A } }  C_  ran  E  <->  { { B ,  B } ,  { B ,  A } }  C_  ran  E ) )
3433notbid 287 . . . . . . . . . . . . . . . . 17  |-  ( x  =  B  ->  ( -.  { { x ,  B } ,  {
x ,  A } }  C_  ran  E  <->  -.  { { B ,  B } ,  { B ,  A } }  C_  ran  E
) )
3529, 34ralprg 3859 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( A. x  e. 
{ A ,  B }  -.  { { x ,  B } ,  {
x ,  A } }  C_  ran  E  <->  ( -.  { { A ,  B } ,  { A ,  A } }  C_  ran  E  /\  -.  { { B ,  B } ,  { B ,  A } }  C_  ran  E
) ) )
3635ad2antrr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( A. x  e.  { A ,  B }  -.  { { x ,  B } ,  { x ,  A } }  C_  ran  E  <->  ( -.  { { A ,  B } ,  { A ,  A } }  C_  ran  E  /\  -.  { { B ,  B } ,  { B ,  A } }  C_  ran  E ) ) )
3724, 36mpbird 225 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  A. x  e.  { A ,  B }  -.  { { x ,  B } ,  {
x ,  A } }  C_  ran  E )
38 ralnex 2717 . . . . . . . . . . . . . 14  |-  ( A. x  e.  { A ,  B }  -.  { { x ,  B } ,  { x ,  A } }  C_  ran  E  <->  -.  E. x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E )
3937, 38sylib 190 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  -.  E. x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E )
4039intnanrd 885 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  -.  ( E. x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  A } }  C_  ran  E  /\  E* x  e.  { A ,  B }  { { x ,  B } ,  { x ,  A } }  C_  ran  E ) )
41 reu5 2923 . . . . . . . . . . . 12  |-  ( E! x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  A } }  C_  ran  E  <->  ( E. x  e.  { A ,  B }  { { x ,  B } ,  { x ,  A } }  C_  ran  E  /\  E* x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E ) )
4240, 41sylnibr 298 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  -.  E! x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  A } }  C_  ran  E
)
432riotaundb 6593 . . . . . . . . . . 11  |-  ( -.  E! x  e.  { A ,  B }  { { x ,  B } ,  { x ,  A } }  C_  ran  E  <->  ( iota_ x  e. 
{ A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) )
4442, 43sylib 190 . . . . . . . . . 10  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( iota_ x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) )
45 preq2 3886 . . . . . . . . . . . . . . . . 17  |-  ( l  =  A  ->  { x ,  l }  =  { x ,  A } )
4645preq2d 3892 . . . . . . . . . . . . . . . 16  |-  ( l  =  A  ->  { {
x ,  B } ,  { x ,  l } }  =  { { x ,  B } ,  { x ,  A } } )
4746sseq1d 3377 . . . . . . . . . . . . . . 15  |-  ( l  =  A  ->  ( { { x ,  B } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  B } ,  {
x ,  A } }  C_  ran  E ) )
4847riotabidv 6553 . . . . . . . . . . . . . 14  |-  ( l  =  A  ->  ( iota_ x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  l } }  C_  ran  E )  =  ( iota_ x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E ) )
4948eqeq1d 2446 . . . . . . . . . . . . 13  |-  ( l  =  A  ->  (
( iota_ x  e.  { A ,  B }  { { x ,  B } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  <->  ( iota_ x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
5049rexsng 3849 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  ( E. l  e.  { A }  ( iota_ x  e. 
{ A ,  B }  { { x ,  B } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
)  <->  ( iota_ x  e. 
{ A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
5150adantr 453 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( E. l  e. 
{ A }  ( iota_ x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B } )  <->  ( iota_ x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
5251ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( E. l  e.  { A }  ( iota_ x  e. 
{ A ,  B }  { { x ,  B } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
)  <->  ( iota_ x  e. 
{ A ,  B }  { { x ,  B } ,  {
x ,  A } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
5344, 52mpbird 225 . . . . . . . . 9  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  E. l  e.  { A }  ( iota_ x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B } ) )
5453olcd 384 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( E. l  e.  { B }  ( iota_ x  e. 
{ A ,  B }  { { x ,  A } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
)  \/  E. l  e.  { A }  ( iota_ x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B } ) ) )
55 difprsn1 3937 . . . . . . . . . . . 12  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B }
)
5655adantl 454 . . . . . . . . . . 11  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B )  ->  ( { A ,  B }  \  { A } )  =  { B }
)
5756rexeqdv 2913 . . . . . . . . . 10  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B )  ->  ( E. l  e.  ( { A ,  B }  \  { A } ) ( iota_ x  e.  { A ,  B }  { { x ,  A } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  <->  E. l  e.  { B }  ( iota_ x  e.  { A ,  B }  { {
x ,  A } ,  { x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B } ) ) )
58 difprsn2 3938 . . . . . . . . . . . 12  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A }
)
5958adantl 454 . . . . . . . . . . 11  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B )  ->  ( { A ,  B }  \  { B } )  =  { A }
)
6059rexeqdv 2913 . . . . . . . . . 10  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B )  ->  ( E. l  e.  ( { A ,  B }  \  { B } ) ( iota_ x  e.  { A ,  B }  { { x ,  B } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  <->  E. l  e.  { A }  ( iota_ x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B } ) ) )
6157, 60orbi12d 692 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B )  ->  ( ( E. l  e.  ( { A ,  B }  \  { A } ) ( iota_ x  e.  { A ,  B }  { { x ,  A } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  \/ 
E. l  e.  ( { A ,  B }  \  { B }
) ( iota_ x  e. 
{ A ,  B }  { { x ,  B } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) )  <->  ( E. l  e.  { B }  ( iota_ x  e. 
{ A ,  B }  { { x ,  A } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
)  \/  E. l  e.  { A }  ( iota_ x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B } ) ) ) )
6261adantr 453 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( ( E. l  e.  ( { A ,  B }  \  { A } ) ( iota_ x  e.  { A ,  B }  { { x ,  A } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  \/ 
E. l  e.  ( { A ,  B }  \  { B }
) ( iota_ x  e. 
{ A ,  B }  { { x ,  B } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) )  <->  ( E. l  e.  { B }  ( iota_ x  e. 
{ A ,  B }  { { x ,  A } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
)  \/  E. l  e.  { A }  ( iota_ x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B } ) ) ) )
6354, 62mpbird 225 . . . . . . 7  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( E. l  e.  ( { A ,  B }  \  { A } ) ( iota_ x  e.  { A ,  B }  { { x ,  A } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  \/ 
E. l  e.  ( { A ,  B }  \  { B }
) ( iota_ x  e. 
{ A ,  B }  { { x ,  B } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
64 sneq 3827 . . . . . . . . . . 11  |-  ( k  =  A  ->  { k }  =  { A } )
6564difeq2d 3467 . . . . . . . . . 10  |-  ( k  =  A  ->  ( { A ,  B }  \  { k } )  =  ( { A ,  B }  \  { A } ) )
66 preq2 3886 . . . . . . . . . . . . . 14  |-  ( k  =  A  ->  { x ,  k }  =  { x ,  A } )
6766preq1d 3891 . . . . . . . . . . . . 13  |-  ( k  =  A  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  A } ,  { x ,  l } }
)
6867sseq1d 3377 . . . . . . . . . . . 12  |-  ( k  =  A  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  A } ,  {
x ,  l } }  C_  ran  E ) )
6968riotabidv 6553 . . . . . . . . . . 11  |-  ( k  =  A  ->  ( iota_ x  e.  { A ,  B }  { {
x ,  k } ,  { x ,  l } }  C_  ran  E )  =  (
iota_ x  e.  { A ,  B }  { {
x ,  A } ,  { x ,  l } }  C_  ran  E ) )
7069eqeq1d 2446 . . . . . . . . . 10  |-  ( k  =  A  ->  (
( iota_ x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  <->  ( iota_ x  e.  { A ,  B }  { { x ,  A } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
7165, 70rexeqbidv 2919 . . . . . . . . 9  |-  ( k  =  A  ->  ( E. l  e.  ( { A ,  B }  \  { k } ) ( iota_ x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  <->  E. l  e.  ( { A ,  B }  \  { A } ) ( iota_ x  e.  { A ,  B }  { { x ,  A } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
72 sneq 3827 . . . . . . . . . . 11  |-  ( k  =  B  ->  { k }  =  { B } )
7372difeq2d 3467 . . . . . . . . . 10  |-  ( k  =  B  ->  ( { A ,  B }  \  { k } )  =  ( { A ,  B }  \  { B } ) )
74 preq2 3886 . . . . . . . . . . . . . 14  |-  ( k  =  B  ->  { x ,  k }  =  { x ,  B } )
7574preq1d 3891 . . . . . . . . . . . . 13  |-  ( k  =  B  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  B } ,  { x ,  l } }
)
7675sseq1d 3377 . . . . . . . . . . . 12  |-  ( k  =  B  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  B } ,  {
x ,  l } }  C_  ran  E ) )
7776riotabidv 6553 . . . . . . . . . . 11  |-  ( k  =  B  ->  ( iota_ x  e.  { A ,  B }  { {
x ,  k } ,  { x ,  l } }  C_  ran  E )  =  (
iota_ x  e.  { A ,  B }  { {
x ,  B } ,  { x ,  l } }  C_  ran  E ) )
7877eqeq1d 2446 . . . . . . . . . 10  |-  ( k  =  B  ->  (
( iota_ x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  <->  ( iota_ x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
7973, 78rexeqbidv 2919 . . . . . . . . 9  |-  ( k  =  B  ->  ( E. l  e.  ( { A ,  B }  \  { k } ) ( iota_ x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  <->  E. l  e.  ( { A ,  B }  \  { B } ) ( iota_ x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
8071, 79rexprg 3860 . . . . . . . 8  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( E. k  e. 
{ A ,  B } E. l  e.  ( { A ,  B }  \  { k } ) ( iota_ x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
)  <->  ( E. l  e.  ( { A ,  B }  \  { A } ) ( iota_ x  e.  { A ,  B }  { { x ,  A } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
)  \/  E. l  e.  ( { A ,  B }  \  { B } ) ( iota_ x  e.  { A ,  B }  { { x ,  B } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) ) )
8180ad2antrr 708 . . . . . . 7  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( E. k  e.  { A ,  B } E. l  e.  ( { A ,  B }  \  { k } ) ( iota_ x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B } )  <->  ( E. l  e.  ( { A ,  B }  \  { A } ) ( iota_ x  e.  { A ,  B }  { { x ,  A } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } )  \/ 
E. l  e.  ( { A ,  B }  \  { B }
) ( iota_ x  e. 
{ A ,  B }  { { x ,  B } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) ) )
8263, 81mpbird 225 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  E. k  e.  { A ,  B } E. l  e.  ( { A ,  B }  \  { k } ) ( iota_ x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) )
83 rexnal 2718 . . . . . . 7  |-  ( E. k  e.  { A ,  B }  -.  A. l  e.  ( { A ,  B }  \  { k } ) E! x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  -.  A. k  e.  { A ,  B } A. l  e.  ( { A ,  B }  \  { k } ) E! x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E )
84 rexnal 2718 . . . . . . . . 9  |-  ( E. l  e.  ( { A ,  B }  \  { k } )  -.  E! x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E  <->  -.  A. l  e.  ( { A ,  B }  \  { k } ) E! x  e.  { A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E )
852riotaundb 6593 . . . . . . . . . . 11  |-  ( -.  E! x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  ( iota_ x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) )
8685a1i 11 . . . . . . . . . 10  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( -.  E! x  e.  { A ,  B }  { {
x ,  k } ,  { x ,  l } }  C_  ran  E  <->  ( iota_ x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
8786rexbidv 2728 . . . . . . . . 9  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( E. l  e.  ( { A ,  B }  \  { k } )  -.  E! x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E  <->  E. l  e.  ( { A ,  B }  \  { k } ) ( iota_ x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B } ) ) )
8884, 87syl5bbr 252 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( -.  A. l  e.  ( { A ,  B }  \  { k } ) E! x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  E. l  e.  ( { A ,  B }  \  { k } ) ( iota_ x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
8988rexbidv 2728 . . . . . . 7  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( E. k  e.  { A ,  B }  -.  A. l  e.  ( { A ,  B }  \  { k } ) E! x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  E. k  e.  { A ,  B } E. l  e.  ( { A ,  B }  \  { k } ) ( iota_ x  e.  { A ,  B }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E )  =  ( Undef `  { A ,  B } ) ) )
9083, 89syl5bbr 252 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( -.  A. k  e.  { A ,  B } A. l  e.  ( { A ,  B }  \  { k } ) E! x  e.  { A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E  <->  E. k  e.  { A ,  B } E. l  e.  ( { A ,  B }  \  { k } ) ( iota_ x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E )  =  ( Undef `  { A ,  B }
) ) )
9182, 90mpbird 225 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  -.  A. k  e.  { A ,  B } A. l  e.  ( { A ,  B }  \  { k } ) E! x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E )
9291intnand 884 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  -.  ( { A ,  B } USGrph  E  /\  A. k  e. 
{ A ,  B } A. l  e.  ( { A ,  B }  \  { k } ) E! x  e. 
{ A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E ) )
93 usgrav 21373 . . . . . 6  |-  ( { A ,  B } USGrph  E  ->  ( { A ,  B }  e.  _V  /\  E  e.  _V )
)
9493adantl 454 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( { A ,  B }  e.  _V  /\  E  e. 
_V ) )
95 isfrgra 28442 . . . . 5  |-  ( ( { A ,  B }  e.  _V  /\  E  e.  _V )  ->  ( { A ,  B } FriendGrph  E  <-> 
( { A ,  B } USGrph  E  /\  A. k  e.  { A ,  B } A. l  e.  ( { A ,  B }  \  { k } ) E! x  e.  { A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E ) ) )
9694, 95syl 16 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  ( { A ,  B } FriendGrph  E  <-> 
( { A ,  B } USGrph  E  /\  A. k  e.  { A ,  B } A. l  e.  ( { A ,  B }  \  { k } ) E! x  e.  { A ,  B }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E ) ) )
9792, 96mtbird 294 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  /\  { A ,  B } USGrph  E )  ->  -.  { A ,  B } FriendGrph  E )
9897expcom 426 . 2  |-  ( { A ,  B } USGrph  E  ->  ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B
)  ->  -.  { A ,  B } FriendGrph  E ) )
99 frisusgra 28444 . . . 4  |-  ( { A ,  B } FriendGrph  E  ->  { A ,  B } USGrph  E )
10099con3i 130 . . 3  |-  ( -. 
{ A ,  B } USGrph  E  ->  -.  { A ,  B } FriendGrph  E )
101100a1d 24 . 2  |-  ( -. 
{ A ,  B } USGrph  E  ->  ( (
( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B )  ->  -.  { A ,  B } FriendGrph  E ) )
10298, 101pm2.61i 159 1  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B )  ->  -.  { A ,  B } FriendGrph  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   E!wreu 2709   E*wrmo 2710   _Vcvv 2958    \ cdif 3319    C_ wss 3322   {csn 3816   {cpr 3817   class class class wbr 4214   ran crn 4881   ` cfv 5456   Undefcund 6543   iota_crio 6544   USGrph cusg 21367   FriendGrph cfrgra 28440
This theorem is referenced by:  1to2vfriswmgra  28458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-hash 11621  df-usgra 21369  df-frgra 28441
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