Theorem rusgrusgr 40764

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 40762	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
2	1	simpld 474	1 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph )

Syntax hints:   → wi 4   ∈ wcel 1977   class class class wbr 4583   USGraph cusgr 40379   RegGraph crgr 40755   RegUSGraph crusgr 40756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rusgr 40758

This theorem is referenced by:  finrusgrfusgr  40765  rusgr0edg  41176  rusgrnumwwlks  41177  rusgrnumwwlk  41178  rusgrnumwlkg  41180  av-numclwwlkovf2num  41516  av-numclwwlk1  41528