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Theorem cusisusgra 21420
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 21419 . 2  |-  ( V ComplUSGrph  E  ->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  E ) )
21simpld 446 1  |-  ( V ComplUSGrph  E  ->  V USGrph  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   A.wral 2666    \ cdif 3277   {csn 3774   {cpr 3775   class class class wbr 4172   ran crn 4838   USGrph cusg 21318   ComplUSGrph ccusgra 21384
This theorem is referenced by:  nbcusgra  21425  cusgrasizeindb0  21432  cusgrasizeindb1  21433  cusgrares  21434  cusgrasizeindslem3  21437  cusgrasizeinds  21438  cusgrasize2inds  21439  cusgrafi  21444  sizeusglecusg  21448  cusconngra  21616
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-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-cusgra 21387
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