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Theorem cusisusgra 24436
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 24435 . 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 1804   A.wral 2793    \ cdif 3458   {csn 4014   {cpr 4016   class class class wbr 4437   ran crn 4990   USGrph cusg 24308   ComplUSGrph ccusgra 24396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-cusgra 24399
This theorem is referenced by:  nbcusgra  24441  cusgrasizeindb0  24448  cusgrasizeindb1  24449  cusgrares  24450  cusgrasizeindslem3  24453  cusgrasizeinds  24454  cusgrasize2inds  24455  cusgrafi  24460  sizeusglecusg  24464  cusconngra  24654  cusgraisrusgra  24916  vdcusgra  32313
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