Theorem rusgrrgr 29596

Description: A k-regular simple graph is a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)

Assertion

Ref	Expression
rusgrrgr	⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 RegGraph 𝐾)

Proof of Theorem rusgrrgr

Step	Hyp	Ref	Expression
1		rusgrprop 29595	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
2	1	simprd 495	1 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 RegGraph 𝐾)

Syntax hints:   → wi 4   ∈ wcel 2106   class class class wbr 5148  USGraphcusgr 29181   RegGraph crgr 29588   RegUSGraph crusgr 29589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rusgr 29591

This theorem is referenced by:  0grrgr  29613  rgrprc  29624  frrusgrord  30370