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Theorem fusgrusgr 40541
 Description: A finite simple graph is a simple graph. (Contributed by AV, 16-Jan-2020.) (Revised by AV, 21-Oct-2020.)
Assertion
Ref Expression
fusgrusgr (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )

Proof of Theorem fusgrusgr
StepHypRef Expression
1 eqid 2610 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
21isfusgr 40537 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3 simpl 472 . 2 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → 𝐺 ∈ USGraph )
42, 3sylbi 206 1 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ‘cfv 5804  Fincfn 7841  Vtxcvtx 25673   USGraph cusgr 40379   FinUSGraph cfusgr 40535 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  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-iota 5768  df-fv 5812  df-fusgr 40536 This theorem is referenced by:  fusgredgfi  40544  fusgrfisstep  40548  nbfiusgrfi  40603  vtxdgfusgrf  40712  usgruvtxvdb  40745  vdiscusgrb  40746  vdiscusgr  40747  fusgrn0eqdrusgr  40770  wlksnfi  41113  fusgrhashclwwlkn  41263  clwlksfclwwlk  41269  clwlksfoclwwlk  41270  clwlksf1clwwlk  41276  fusgr2wsp2nb  41498  fusgreghash2wspv  41499  av-numclwwlk4  41540
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