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Theorem rusgrusgr 26691
 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 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)

Proof of Theorem rusgrusgr
StepHypRef Expression
1 rusgrprop 26689 . 2 (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾))
21simpld 477 1 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2139   class class class wbr 4804  USGraphcusgr 26264  RegGraphcrgr 26682  RegUSGraphcrusgr 26683 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-rusgr 26685 This theorem is referenced by:  finrusgrfusgr  26692  rusgr0edg  27116  rusgrnumwwlks  27117  rusgrnumwwlk  27118  rusgrnumwlkg  27120  numclwwlk1  27541  clwlknon2num  27550  numclwlk1lem1  27551  numclwlk1lem2  27552
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