Theorem rusgrrgr 29650

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 29649	. 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 2114   class class class wbr 5086  USGraphcusgr 29235   RegGraph crgr 29642   RegUSGraph crusgr 29643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rusgr 29645

This theorem is referenced by:  0grrgr  29667  rgrprc  29678  frrusgrord  30429