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Theorem 4cyclusnfrgra 30616
Description: A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Assertion
Ref Expression
4cyclusnfrgra  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D
) )  ->  (
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  -.  V FriendGrph  E ) )

Proof of Theorem 4cyclusnfrgra
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simprl 755 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C
)  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/= 
D ) )  /\  ( ( { 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 ) )
2 simprr 756 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C
)  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/= 
D ) )  /\  ( ( { 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 ) )
3 simpl3 993 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C
)  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/= 
D ) )  /\  ( ( { 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 ) )
4 4cycl2vnunb 30614 . . . . 5  |-  ( ( ( { 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 )
51, 2, 3, 4syl3anc 1218 . . . 4  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C
)  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/= 
D ) )  /\  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) )  ->  -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E
)
6 frgraunss 30592 . . . . . . . 8  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E
) )
7 pm2.24 109 . . . . . . . 8  |-  ( E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E  ->  ( -.  E! x  e.  V  { { A ,  x } ,  {
x ,  C } }  C_  ran  E  ->  -.  V FriendGrph  E ) )
86, 7syl6com 35 . . . . . . 7  |-  ( ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  -> 
( V FriendGrph  E  ->  ( -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E  ->  -.  V FriendGrph  E ) ) )
983ad2ant2 1010 . . . . . 6  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D
) )  ->  ( V FriendGrph  E  ->  ( -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E  ->  -.  V FriendGrph  E ) ) )
109com23 78 . . . . 5  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D
) )  ->  ( -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E  ->  ( V FriendGrph  E  ->  -.  V FriendGrph  E ) ) )
1110adantr 465 . . . 4  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C
)  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/= 
D ) )  /\  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) )  ->  ( -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E  ->  ( V FriendGrph  E  ->  -.  V FriendGrph  E ) ) )
125, 11mpd 15 . . 3  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C
)  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/= 
D ) )  /\  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) )  ->  ( V FriendGrph  E  ->  -.  V FriendGrph  E ) )
1312pm2.01d 169 . 2  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C
)  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/= 
D ) )  /\  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) )  ->  -.  V FriendGrph  E )
1413ex 434 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D
) )  ->  (
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  -.  V FriendGrph  E ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1756    =/= wne 2611   E!wreu 2722    C_ wss 3333   {cpr 3884   class class class wbr 4297   ran crn 4846   USGrph cusg 23269   FriendGrph cfrgra 30585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-frgra 30586
This theorem is referenced by:  frgranbnb  30617  frgrawopreg  30647
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