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Theorem rusgrusgr 27352
Description: A k-regular simple graph is a simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
Assertion
Ref Expression
rusgrusgr (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)

Proof of Theorem rusgrusgr
StepHypRef Expression
1 rusgrprop 27350 . 2 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
21simpld 498 1 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114   class class class wbr 5042  USGraphcusgr 26940   RegGraph crgr 27343   RegUSGraph crusgr 27344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rusgr 27346
This theorem is referenced by:  finrusgrfusgr  27353  rusgr0edg  27757  rusgrnumwwlks  27758  rusgrnumwwlk  27759  rusgrnumwlkg  27761  numclwwlk1  28144  clwlknon2num  28151  numclwlk1lem1  28152  numclwlk1lem2  28153
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