Theorem rusgrusgr 29596

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 𝐾 → 𝐺 ∈ USGraph)

Proof of Theorem rusgrusgr

Step	Hyp	Ref	Expression
1		rusgrprop 29594	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
2	1	simpld 494	1 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2105   class class class wbr 5147  USGraphcusgr 29180   RegGraph crgr 29587   RegUSGraph crusgr 29588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rusgr 29590

This theorem is referenced by:  finrusgrfusgr  29597  rusgr0edg  30002  rusgrnumwwlks  30003  rusgrnumwwlk  30004  rusgrnumwlkg  30006  numclwwlk1  30389  clwlknon2num  30396  numclwlk1lem1  30397  numclwlk1lem2  30398