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Theorem cusisusgra 24134
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 24133 . 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 1767   A.wral 2814    \ cdif 3473   {csn 4027   {cpr 4029   class class class wbr 4447   ran crn 5000   USGrph cusg 24006   ComplUSGrph ccusgra 24094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-cusgra 24097
This theorem is referenced by:  nbcusgra  24139  cusgrasizeindb0  24146  cusgrasizeindb1  24147  cusgrares  24148  cusgrasizeindslem3  24151  cusgrasizeinds  24152  cusgrasize2inds  24153  cusgrafi  24158  sizeusglecusg  24162  cusconngra  24352  cusgraisrusgra  24614  vdcusgra  31828
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