Theorem rusgrusgr 27834

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 27832	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
2	1	simpld 494	1 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2108   class class class wbr 5070  USGraphcusgr 27422   RegGraph crgr 27825   RegUSGraph crusgr 27826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rusgr 27828

This theorem is referenced by:  finrusgrfusgr  27835  rusgr0edg  28239  rusgrnumwwlks  28240  rusgrnumwwlk  28241  rusgrnumwlkg  28243  numclwwlk1  28626  clwlknon2num  28633  numclwlk1lem1  28634  numclwlk1lem2  28635