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Theorem 4cyclusnfrgra 25826
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 772 . . . . 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 774 . . . . 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 1035 . . . . 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 25824 . . . . 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 1292 . . . 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 25802 . . . . . . . 8  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E
) )
7 pm2.24 112 . . . . . . . 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 1052 . . . . . 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 80 . . . . 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 472 . . . 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 174 . 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 441 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 376    /\ w3a 1007    e. wcel 1904    =/= wne 2641   E!wreu 2758    C_ wss 3390   {cpr 3961   class class class wbr 4395   ran crn 4840   USGrph cusg 25136   FriendGrph cfrgra 25795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-frgra 25796
This theorem is referenced by:  frgranbnb  25827  frgrawopreg  25856
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