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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 USGrph FriendGrph

Proof of Theorem 4cyclusnfrgra
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simprl 772 . . . . 5 USGrph
2 simprr 774 . . . . 5 USGrph
3 simpl3 1035 . . . . 5 USGrph
4 4cycl2vnunb 25824 . . . . 5
51, 2, 3, 4syl3anc 1292 . . . 4 USGrph
6 frgraunss 25802 . . . . . . . 8 FriendGrph
7 pm2.24 112 . . . . . . . 8 FriendGrph
86, 7syl6com 35 . . . . . . 7 FriendGrph FriendGrph
983ad2ant2 1052 . . . . . 6 USGrph FriendGrph FriendGrph
109com23 80 . . . . 5 USGrph FriendGrph FriendGrph
1110adantr 472 . . . 4 USGrph FriendGrph FriendGrph
125, 11mpd 15 . . 3 USGrph FriendGrph FriendGrph
1312pm2.01d 174 . 2 USGrph FriendGrph
1413ex 441 1 USGrph FriendGrph
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 376   w3a 1007   wcel 1904   wne 2641  wreu 2758   wss 3390  cpr 3961   class class class wbr 4395   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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