MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  n4cyclfrgra Structured version   Unicode version

Theorem n4cyclfrgra 25144
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 25118 . . . 4  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 4cycl4dv4e 24794 . . . . . . . 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 25117 . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( c  e.  V  /\  d  e.  V )  ->  c  e.  V )
54adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
c  e.  V )
65adantr 465 . . . . . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  =/=  c  <->  c  =/=  a )
87biimpi 194 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =/=  c  ->  c  =/=  a )
983ad2ant2 1018 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  ->  c  =/=  a )
109ad2antrl 727 . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . 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 4157 . . . . . . . . . . . . . . . . . 18  |-  ( c  e.  ( V  \  { a } )  <-> 
( c  e.  V  /\  c  =/=  a
) )
136, 11, 12sylanbrc 664 . . . . . . . . . . . . . . . . 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 4042 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  a  ->  { k }  =  { a } )
1514difeq2d 3618 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  a  ->  ( V  \  { k } )  =  ( V 
\  { a } ) )
16 preq2 4112 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  a  ->  { x ,  k }  =  { x ,  a } )
1716preq1d 4117 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  a  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  a } ,  { x ,  l } }
)
1817sseq1d 3526 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  a  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  a } ,  {
x ,  l } }  C_  ran  E ) )
1918reubidv 3042 . . . . . . . . . . . . . . . . . . . . . 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 3068 . . . . . . . . . . . . . . . . . . . . 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 3206 . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 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 4112 . . . . . . . . . . . . . . . . . . . . 21  |-  ( l  =  c  ->  { x ,  l }  =  { x ,  c } )
2625preq2d 4118 . . . . . . . . . . . . . . . . . . . 20  |-  ( l  =  c  ->  { {
x ,  a } ,  { x ,  l } }  =  { { x ,  a } ,  { x ,  c } }
)
2726sseq1d 3526 . . . . . . . . . . . . . . . . . . 19  |-  ( l  =  c  ->  ( { { x ,  a } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  a } ,  {
x ,  c } }  C_  ran  E ) )
2827reubidv 3042 . . . . . . . . . . . . . . . . . 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 3206 . . . . . . . . . . . . . . . . 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 56 . . . . . . . . . . . . . . . 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 4110 . . . . . . . . . . . . . . . . . . . 20  |-  { x ,  a }  =  { a ,  x }
3231preq1i 4114 . . . . . . . . . . . . . . . . . . 19  |-  { {
x ,  a } ,  { x ,  c } }  =  { { a ,  x } ,  { x ,  c } }
3332sseq1i 3523 . . . . . . . . . . . . . . . . . 18  |-  ( { { x ,  a } ,  { x ,  c } }  C_ 
ran  E  <->  { { a ,  x } ,  {
x ,  c } }  C_  ran  E )
3433reubii 3044 . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  V  /\  b  e.  V )  ->  b  e.  V )
4039adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
b  e.  V )
4140adantr 465 . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( c  e.  V  /\  d  e.  V )  ->  d  e.  V )
4342adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
d  e.  V )
4443adantr 465 . . . . . . . . . . . . . . . . . . . 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 1045 . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . 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 25143 . . . . . . . . . . . . . . . . . . . 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 1240 . . . . . . . . . . . . . . . . . . 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 106 . . . . . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . . . . . 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 195 . . . . . . . . . . . . . . . 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 33 . . . . . . . . . . . . . . 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 50 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 31 . . . . . . . . . . 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 434 . . . . . . . . . 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 2956 . . . . . . . . 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 2954 . . . . . . . 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 1195 . . . . . 6  |-  ( V USGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( ( # `  F )  =  4  ->  ( V FriendGrph  E  -> 
( # `  F )  =/=  4 ) ) ) )
6261com34 83 . . . . 5  |-  ( V USGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( V FriendGrph  E  -> 
( ( # `  F
)  =  4  -> 
( # `  F )  =/=  4 ) ) ) )
6362com23 78 . . . 4  |-  ( V USGrph  E  ->  ( V FriendGrph  E  -> 
( F ( V Cycles  E ) P  -> 
( ( # `  F
)  =  4  -> 
( # `  F )  =/=  4 ) ) ) )
641, 63mpcom 36 . . 3  |-  ( V FriendGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( ( # `  F )  =  4  ->  ( # `  F
)  =/=  4 ) ) )
6564imp 429 . 2  |-  ( ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( ( # `
 F )  =  4  ->  ( # `  F
)  =/=  4 ) )
66 df-ne 2654 . . 3  |-  ( (
# `  F )  =/=  4  <->  -.  ( # `  F
)  =  4 )
6766biimpri 206 . 2  |-  ( -.  ( # `  F
)  =  4  -> 
( # `  F )  =/=  4 )
6865, 67pm2.61d1 159 1  |-  ( ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `  F
)  =/=  4 )
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.wrex 2808   E!wreu 2809    \ cdif 3468    C_ wss 3471   {csn 4032   {cpr 4034   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   4c4 10608   #chash 12407   USGrph cusg 24456   Cycles ccycl 24633   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-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-usgra 24459  df-wlk 24634  df-trail 24635  df-pth 24636  df-cycl 24639  df-frgra 25115
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator