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Theorem rusisusgra 26458
 Description: Any k-regular undirected simple graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
rusisusgra (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)

Proof of Theorem rusisusgra
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 rusgraprop 26456 . 2 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))
21simp1d 1066 1 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℕ0cn0 11169   USGrph cusg 25859   VDeg cvdg 26420   RegUSGrph crusgra 26450 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  rusgranumwwlkl1  26473  rusgranumwlkl1  26474  rusgranumwlkb1  26481  rusgra0edg  26482  rusgranumwlks  26483  rusgranumwlk  26484  rusgranumwwlkg  26486  numclwwlkovf2num  26612  numclwwlk1  26625  numclwwlkqhash  26627  numclwwlk3  26636  numclwwlk5  26639  numclwwlk6  26640  frgrareg  26644
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