Theorem rusgrusgr 29549

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 29547	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
2	1	simpld 494	1 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2109   class class class wbr 5124  USGraphcusgr 29133   RegGraph crgr 29540   RegUSGraph crusgr 29541

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rusgr 29543

This theorem is referenced by:  finrusgrfusgr  29550  rusgr0edg  29960  rusgrnumwwlks  29961  rusgrnumwwlk  29962  rusgrnumwlkg  29964  numclwwlk1  30347  clwlknon2num  30354  numclwlk1lem1  30355  numclwlk1lem2  30356