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Theorem 4cyclusnfrgra 28123
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 733 . . . . 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 734 . . . . 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 962 . . . . 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 28121 . . . . 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 1184 . . . 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 28099 . . . . . . . 8  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E
) )
7 pm2.24 103 . . . . . . . 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 33 . . . . . . 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 979 . . . . . 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 74 . . . . 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 452 . . . 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 163 . 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 424 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721    =/= wne 2567   E!wreu 2668    C_ wss 3280   {cpr 3775   class class class wbr 4172   ran crn 4838   USGrph cusg 21318   FriendGrph cfrgra 28092
This theorem is referenced by:  frgranbnb  28124  frgrawopreg  28152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-frgra 28093
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