Theorem rusgrusgr 29600

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 29598	. 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 5166  USGraphcusgr 29184   RegGraph crgr 29591   RegUSGraph crusgr 29592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rusgr 29594

This theorem is referenced by:  finrusgrfusgr  29601  rusgr0edg  30006  rusgrnumwwlks  30007  rusgrnumwwlk  30008  rusgrnumwlkg  30010  numclwwlk1  30393  clwlknon2num  30400  numclwlk1lem1  30401  numclwlk1lem2  30402