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Theorem rusgrusgr 29582
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 29580 . 2 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
21simpld 494 1 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108   class class class wbr 5143  USGraphcusgr 29166   RegGraph crgr 29573   RegUSGraph crusgr 29574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rusgr 29576
This theorem is referenced by:  finrusgrfusgr  29583  rusgr0edg  29993  rusgrnumwwlks  29994  rusgrnumwwlk  29995  rusgrnumwlkg  29997  numclwwlk1  30380  clwlknon2num  30387  numclwlk1lem1  30388  numclwlk1lem2  30389
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