Theorem rusgrusgr 30127

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 30125	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
2	1	simpld 500	1 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   class class class wbr 5103  USGraphcusgr 29712   RegGraph crgr 30118   RegUSGraph crusgr 30119

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rusgr 30121

This theorem is used by:  finrusgrfusgr  30128  rusgr0edg  30547  rusgrnumwwlks  30548  rusgrnumwwlk  30549  rusgrnumwlkg  30551  numclwwlk1  30944  clwlknon2num  30951  numclwlk1lem1  30952  numclwlk1lem2  30953