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Theorem finrusgrfusgr 28802
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 28801 . . 3 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
21anim1i 616 . 2 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 finrusgrfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
43isfusgr 28555 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
52, 4sylibr 233 1 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107   class class class wbr 5147  cfv 6540  Fincfn 8935  Vtxcvtx 28236  USGraphcusgr 28389  FinUSGraphcfusgr 28553   RegUSGraph crusgr 28793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-iota 6492  df-fv 6548  df-fusgr 28554  df-rusgr 28795
This theorem is referenced by:  numclwwlk1  29594  numclwwlk3  29618  numclwwlk5  29621  numclwwlk7lem  29622  numclwwlk6  29623  frgrreggt1  29626
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