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Theorem n4cyclfrgra 28481
Description: There is no 4-cycle in a friendship graph, see Proposition 1 of [MertziosUnger] p. 153 : "A friendship graph G contains no C4 as a subgraph ...". (Contributed by Alexander van der Vekens, 19-Nov-2017.)
Assertion
Ref Expression
n4cyclfrgra  |-  ( ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `  F
)  =/=  4 )

Proof of Theorem n4cyclfrgra
Dummy variables  a 
b  c  d  k  l  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 28455 . . . 4  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 4cycl4dv4e 21660 . . . . . . . 8  |-  ( ( V USGrph  E  /\  F ( V Cycles  E ) P  /\  ( # `  F
)  =  4 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )
3 frisusgrapr 28454 . . . . . . . . . . . . 13  |-  ( V FriendGrph  E  ->  ( V USGrph  E  /\  A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  E ) )
4 simpl 445 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( c  e.  V  /\  d  e.  V )  ->  c  e.  V )
54adantl 454 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
c  e.  V )
65adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  c  e.  V )
7 necom 2687 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  =/=  c  <->  c  =/=  a )
87biimpi 188 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =/=  c  ->  c  =/=  a )
983ad2ant2 980 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  ->  c  =/=  a )
109ad2antrl 710 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E
) )  /\  (
( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d
)  /\  ( b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) )  ->  c  =/=  a
)
1110adantl 454 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  c  =/=  a )
12 eldifsn 3929 . . . . . . . . . . . . . . . . . 18  |-  ( c  e.  ( V  \  { a } )  <-> 
( c  e.  V  /\  c  =/=  a
) )
136, 11, 12sylanbrc 647 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  c  e.  ( V  \  { a } ) )
14 sneq 3827 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  a  ->  { k }  =  { a } )
1514difeq2d 3467 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  a  ->  ( V  \  { k } )  =  ( V 
\  { a } ) )
16 preq2 3886 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  a  ->  { x ,  k }  =  { x ,  a } )
1716preq1d 3891 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  a  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  a } ,  { x ,  l } }
)
1817sseq1d 3377 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  a  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  a } ,  {
x ,  l } }  C_  ran  E ) )
1918reubidv 2894 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  a  ->  ( E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  E! x  e.  V  { { x ,  a } ,  { x ,  l } }  C_ 
ran  E ) )
2015, 19raleqbidv 2918 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  a  ->  ( A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  E  <->  A. l  e.  ( V  \  { a } ) E! x  e.  V  { { x ,  a } ,  { x ,  l } }  C_  ran  E ) )
2120rspcv 3050 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  e.  V  ->  ( A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  E  ->  A. l  e.  ( V  \  { a } ) E! x  e.  V  { { x ,  a } ,  { x ,  l } }  C_  ran  E ) )
2221adantr 453 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  e.  V  /\  b  e.  V )  ->  ( A. k  e.  V  A. l  e.  ( V  \  {
k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  ->  A. l  e.  ( V  \  {
a } ) E! x  e.  V  { { x ,  a } ,  { x ,  l } }  C_ 
ran  E ) )
2322adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
( A. k  e.  V  A. l  e.  ( V  \  {
k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  ->  A. l  e.  ( V  \  {
a } ) E! x  e.  V  { { x ,  a } ,  { x ,  l } }  C_ 
ran  E ) )
2423adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  ->  A. l  e.  ( V  \  {
a } ) E! x  e.  V  { { x ,  a } ,  { x ,  l } }  C_ 
ran  E ) )
25 preq2 3886 . . . . . . . . . . . . . . . . . . . . 21  |-  ( l  =  c  ->  { x ,  l }  =  { x ,  c } )
2625preq2d 3892 . . . . . . . . . . . . . . . . . . . 20  |-  ( l  =  c  ->  { {
x ,  a } ,  { x ,  l } }  =  { { x ,  a } ,  { x ,  c } }
)
2726sseq1d 3377 . . . . . . . . . . . . . . . . . . 19  |-  ( l  =  c  ->  ( { { x ,  a } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  a } ,  {
x ,  c } }  C_  ran  E ) )
2827reubidv 2894 . . . . . . . . . . . . . . . . . 18  |-  ( l  =  c  ->  ( E! x  e.  V  { { x ,  a } ,  { x ,  l } }  C_ 
ran  E  <->  E! x  e.  V  { { x ,  a } ,  { x ,  c } }  C_ 
ran  E ) )
2928rspcv 3050 . . . . . . . . . . . . . . . . 17  |-  ( c  e.  ( V  \  { a } )  ->  ( A. l  e.  ( V  \  {
a } ) E! x  e.  V  { { x ,  a } ,  { x ,  l } }  C_ 
ran  E  ->  E! x  e.  V  { { x ,  a } ,  { x ,  c } }  C_  ran  E ) )
3013, 24, 29sylsyld 55 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  ->  E! x  e.  V  { { x ,  a } ,  { x ,  c } }  C_  ran  E ) )
31 prcom 3884 . . . . . . . . . . . . . . . . . . . 20  |-  { x ,  a }  =  { a ,  x }
3231preq1i 3888 . . . . . . . . . . . . . . . . . . 19  |-  { {
x ,  a } ,  { x ,  c } }  =  { { a ,  x } ,  { x ,  c } }
3332sseq1i 3374 . . . . . . . . . . . . . . . . . 18  |-  ( { { x ,  a } ,  { x ,  c } }  C_ 
ran  E  <->  { { a ,  x } ,  {
x ,  c } }  C_  ran  E )
3433reubii 2896 . . . . . . . . . . . . . . . . 17  |-  ( E! x  e.  V  { { x ,  a } ,  { x ,  c } }  C_ 
ran  E  <->  E! x  e.  V  { { a ,  x } ,  { x ,  c } }  C_ 
ran  E )
35 simpl 445 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  ->  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
3635ad2antrl 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
37 simpr 449 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  ->  ( {
c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )
3837ad2antrl 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( {
c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )
39 simpr 449 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  V  /\  b  e.  V )  ->  b  e.  V )
4039adantr 453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
b  e.  V )
4140adantr 453 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  b  e.  V )
42 simpr 449 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( c  e.  V  /\  d  e.  V )  ->  d  e.  V )
4342adantl 454 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
d  e.  V )
4443adantr 453 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  d  e.  V )
45 simprr2 1007 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E
) )  /\  (
( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d
)  /\  ( b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) )  ->  b  =/=  d
)
4645adantl 454 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  b  =/=  d )
47 4cycl2vnunb 28480 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E )  /\  ( b  e.  V  /\  d  e.  V  /\  b  =/=  d ) )  ->  -.  E! x  e.  V  { { a ,  x } ,  { x ,  c } }  C_ 
ran  E )
4836, 38, 41, 44, 46, 47syl113anc 1197 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  -.  E! x  e.  V  { { a ,  x } ,  { x ,  c } }  C_ 
ran  E )
4948pm2.21d 101 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( E! x  e.  V  { { a ,  x } ,  { x ,  c } }  C_ 
ran  E  ->  ( # `  F )  =/=  4
) )
5049com12 30 . . . . . . . . . . . . . . . . 17  |-  ( E! x  e.  V  { { a ,  x } ,  { x ,  c } }  C_ 
ran  E  ->  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( # `  F
)  =/=  4 ) )
5134, 50sylbi 189 . . . . . . . . . . . . . . . 16  |-  ( E! x  e.  V  { { x ,  a } ,  { x ,  c } }  C_ 
ran  E  ->  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( # `  F
)  =/=  4 ) )
5230, 51syl6 32 . . . . . . . . . . . . . . 15  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  ->  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( # `  F
)  =/=  4 ) ) )
5352pm2.43b 49 . . . . . . . . . . . . . 14  |-  ( A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  ->  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( # `  F
)  =/=  4 ) )
5453adantl 454 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  A. k  e.  V  A. l  e.  ( V  \  {
k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E )  -> 
( ( ( ( a  e.  V  /\  b  e.  V )  /\  ( c  e.  V  /\  d  e.  V
) )  /\  (
( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E
) )  /\  (
( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d
)  /\  ( b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( # `  F
)  =/=  4 ) )
553, 54syl 16 . . . . . . . . . . . 12  |-  ( V FriendGrph  E  ->  ( ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  /\  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( # `  F
)  =/=  4 ) )
5655com12 30 . . . . . . . . . . 11  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  (
c  e.  V  /\  d  e.  V )
)  /\  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  ->  ( V FriendGrph  E  ->  ( # `  F
)  =/=  4 ) )
5756ex 425 . . . . . . . . . 10  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
( ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  /\  ( {
c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) )  ->  ( V FriendGrph  E  -> 
( # `  F )  =/=  4 ) ) )
5857rexlimdvva 2839 . . . . . . . . 9  |-  ( ( a  e.  V  /\  b  e.  V )  ->  ( E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  /\  ( {
c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) )  ->  ( V FriendGrph  E  -> 
( # `  F )  =/=  4 ) ) )
5958rexlimivv 2837 . . . . . . . 8  |-  ( E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  (
( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E
) )  /\  (
( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d
)  /\  ( b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) )  ->  ( V FriendGrph  E  -> 
( # `  F )  =/=  4 ) )
602, 59syl 16 . . . . . . 7  |-  ( ( V USGrph  E  /\  F ( V Cycles  E ) P  /\  ( # `  F
)  =  4 )  ->  ( V FriendGrph  E  -> 
( # `  F )  =/=  4 ) )
61603exp 1153 . . . . . 6  |-  ( V USGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( ( # `  F )  =  4  ->  ( V FriendGrph  E  -> 
( # `  F )  =/=  4 ) ) ) )
6261com34 80 . . . . 5  |-  ( V USGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( V FriendGrph  E  -> 
( ( # `  F
)  =  4  -> 
( # `  F )  =/=  4 ) ) ) )
6362com23 75 . . . 4  |-  ( V USGrph  E  ->  ( V FriendGrph  E  -> 
( F ( V Cycles  E ) P  -> 
( ( # `  F
)  =  4  -> 
( # `  F )  =/=  4 ) ) ) )
641, 63mpcom 35 . . 3  |-  ( V FriendGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( ( # `  F )  =  4  ->  ( # `  F
)  =/=  4 ) ) )
6564imp 420 . 2  |-  ( ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( ( # `
 F )  =  4  ->  ( # `  F
)  =/=  4 ) )
66 df-ne 2603 . . 3  |-  ( (
# `  F )  =/=  4  <->  -.  ( # `  F
)  =  4 )
6766biimpri 199 . 2  |-  ( -.  ( # `  F
)  =  4  -> 
( # `  F )  =/=  4 )
6865, 67pm2.61d1 154 1  |-  ( ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `  F
)  =/=  4 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   E!wreu 2709    \ cdif 3319    C_ wss 3322   {csn 3816   {cpr 3817   class class class wbr 4215   ran crn 4882   ` cfv 5457  (class class class)co 6084   4c4 10056   #chash 11623   USGrph cusg 21370   Cycles ccycl 21520   FriendGrph cfrgra 28451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
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 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-hash 11624  df-word 11728  df-usgra 21372  df-wlk 21521  df-trail 21522  df-pth 21523  df-cycl 21526  df-frgra 28452
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