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Theorem finrusgrfusgr 29598
Description: A finite regular simple graph is a finite simple graph. (Contributed by AV, 3-Jun-2021.)
Hypothesis
Ref Expression
finrusgrfusgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
finrusgrfusgr ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)

Proof of Theorem finrusgrfusgr
StepHypRef Expression
1 rusgrusgr 29597 . . 3 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
21anim1i 615 . 2 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 finrusgrfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
43isfusgr 29350 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
52, 4sylibr 234 1 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  Fincfn 8984  Vtxcvtx 29028  USGraphcusgr 29181  FinUSGraphcfusgr 29348   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-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-iota 6516  df-fv 6571  df-fusgr 29349  df-rusgr 29591
This theorem is referenced by:  numclwwlk1  30390  numclwwlk3  30414  numclwwlk5  30417  numclwwlk7lem  30418  numclwwlk6  30419  frgrreggt1  30422
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