Theorem rusgrrgr 29467

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 29466	. 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 2109   class class class wbr 5102  USGraphcusgr 29052   RegGraph crgr 29459   RegUSGraph crusgr 29460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rusgr 29462

This theorem is referenced by:  0grrgr  29484  rgrprc  29495  frrusgrord  30243