Theorem rusgrusgr 30178

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 30176	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
2	1	simpld 500	1 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   class class class wbr 5106  USGraphcusgr 29763   RegGraph crgr 30169   RegUSGraph crusgr 30170

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rusgr 30172

This theorem is used by:  finrusgrfusgr  30179  rusgr0edg  30598  rusgrnumwwlks  30599  rusgrnumwwlk  30600  rusgrnumwlkg  30602  numclwwlk1  30995  clwlknon2num  31002  numclwlk1lem1  31003  numclwlk1lem2  31004