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Theorem n4cyclfrgra 25746
Description: There is no 4-cycle in a friendship graph, see Proposition 1(a) 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 25720 . . . 4  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 4cycl4dv4e 25396 . . . . . . . 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 25719 . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( c  e.  V  /\  d  e.  V )  ->  c  e.  V )
54adantl 468 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
c  e.  V )
65adantr 467 . . . . . . . . . . . . . . . . . 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 2677 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  =/=  c  <->  c  =/=  a )
87biimpi 198 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =/=  c  ->  c  =/=  a )
983ad2ant2 1030 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  ->  c  =/=  a )
109ad2antrl 734 . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . 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 4097 . . . . . . . . . . . . . . . . . 18  |-  ( c  e.  ( V  \  { a } )  <-> 
( c  e.  V  /\  c  =/=  a
) )
136, 11, 12sylanbrc 670 . . . . . . . . . . . . . . . . 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 3978 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  a  ->  { k }  =  { a } )
1514difeq2d 3551 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  a  ->  ( V  \  { k } )  =  ( V 
\  { a } ) )
16 preq2 4052 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  a  ->  { x ,  k }  =  { x ,  a } )
1716preq1d 4057 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  a  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  a } ,  { x ,  l } }
)
1817sseq1d 3459 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  a  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  a } ,  {
x ,  l } }  C_  ran  E ) )
1918reubidv 2975 . . . . . . . . . . . . . . . . . . . . . 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 3001 . . . . . . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 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 4052 . . . . . . . . . . . . . . . . . . . . 21  |-  ( l  =  c  ->  { x ,  l }  =  { x ,  c } )
2625preq2d 4058 . . . . . . . . . . . . . . . . . . . 20  |-  ( l  =  c  ->  { {
x ,  a } ,  { x ,  l } }  =  { { x ,  a } ,  { x ,  c } }
)
2726sseq1d 3459 . . . . . . . . . . . . . . . . . . 19  |-  ( l  =  c  ->  ( { { x ,  a } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  a } ,  {
x ,  c } }  C_  ran  E ) )
2827reubidv 2975 . . . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . . . . . 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 58 . . . . . . . . . . . . . . . 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 4050 . . . . . . . . . . . . . . . . . . . 20  |-  { x ,  a }  =  { a ,  x }
3231preq1i 4054 . . . . . . . . . . . . . . . . . . 19  |-  { {
x ,  a } ,  { x ,  c } }  =  { { a ,  x } ,  { x ,  c } }
3332sseq1i 3456 . . . . . . . . . . . . . . . . . 18  |-  ( { { x ,  a } ,  { x ,  c } }  C_ 
ran  E  <->  { { a ,  x } ,  {
x ,  c } }  C_  ran  E )
3433reubii 2977 . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  V  /\  b  e.  V )  ->  b  e.  V )
4039adantr 467 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
b  e.  V )
4140adantr 467 . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( c  e.  V  /\  d  e.  V )  ->  d  e.  V )
4342adantl 468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
d  e.  V )
4443adantr 467 . . . . . . . . . . . . . . . . . . . 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 1057 . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . 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 25745 . . . . . . . . . . . . . . . . . . . 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 1280 . . . . . . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . . . . 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 199 . . . . . . . . . . . . . . . 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 34 . . . . . . . . . . . . . . 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 52 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . 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 17 . . . . . . . . . . . 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 32 . . . . . . . . . . 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 436 . . . . . . . . . 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 2886 . . . . . . . . 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 2884 . . . . . . . 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 17 . . . . . . 7  |-  ( ( V USGrph  E  /\  F ( V Cycles  E ) P  /\  ( # `  F
)  =  4 )  ->  ( V FriendGrph  E  -> 
( # `  F )  =/=  4 ) )
61603exp 1207 . . . . . 6  |-  ( V USGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( ( # `  F )  =  4  ->  ( V FriendGrph  E  -> 
( # `  F )  =/=  4 ) ) ) )
6261com34 86 . . . . 5  |-  ( V USGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( V FriendGrph  E  -> 
( ( # `  F
)  =  4  -> 
( # `  F )  =/=  4 ) ) ) )
6362com23 81 . . . 4  |-  ( V USGrph  E  ->  ( V FriendGrph  E  -> 
( F ( V Cycles  E ) P  -> 
( ( # `  F
)  =  4  -> 
( # `  F )  =/=  4 ) ) ) )
641, 63mpcom 37 . . 3  |-  ( V FriendGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( ( # `  F )  =  4  ->  ( # `  F
)  =/=  4 ) ) )
6564imp 431 . 2  |-  ( ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( ( # `
 F )  =  4  ->  ( # `  F
)  =/=  4 ) )
66 df-ne 2624 . . 3  |-  ( (
# `  F )  =/=  4  <->  -.  ( # `  F
)  =  4 )
6766biimpri 210 . 2  |-  ( -.  ( # `  F
)  =  4  -> 
( # `  F )  =/=  4 )
6865, 67pm2.61d1 163 1  |-  ( ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `  F
)  =/=  4 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   E!wreu 2739    \ cdif 3401    C_ wss 3404   {csn 3968   {cpr 3970   class class class wbr 4402   ran crn 4835   ` cfv 5582  (class class class)co 6290   4c4 10661   #chash 12515   USGrph cusg 25057   Cycles ccycl 25235   FriendGrph cfrgra 25716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-usgra 25060  df-wlk 25236  df-trail 25237  df-pth 25238  df-cycl 25241  df-frgra 25717
This theorem is referenced by: (None)
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