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Theorem cusisusgra 23371
Description: A complete (undirected simple) graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
Assertion
Ref Expression
cusisusgra  |-  ( V ComplUSGrph  E  ->  V USGrph  E )

Proof of Theorem cusisusgra
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscusgra0 23370 . 2  |-  ( V ComplUSGrph  E  ->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  E ) )
21simpld 459 1  |-  ( V ComplUSGrph  E  ->  V USGrph  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   A.wral 2720    \ cdif 3330   {csn 3882   {cpr 3884   class class class wbr 4297   ran crn 4846   USGrph cusg 23269   ComplUSGrph ccusgra 23335
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-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-cusgra 23338
This theorem is referenced by:  nbcusgra  23376  cusgrasizeindb0  23383  cusgrasizeindb1  23384  cusgrares  23385  cusgrasizeindslem3  23388  cusgrasizeinds  23389  cusgrasize2inds  23390  cusgrafi  23395  sizeusglecusg  23399  cusconngra  23567  vdcusgra  30536  cusgraisrusgra  30556
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