Theorem fusgrusgr 27670

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

Step	Hyp	Ref	Expression
1		eqid 2739	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 27666	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2109  ‘cfv 6430  Fincfn 8707  Vtxcvtx 27347  USGraphcusgr 27500  FinUSGraphcfusgr 27664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-fusgr 27665

This theorem is referenced by:  fusgredgfi  27673  fusgrfisstep  27677  fusgrfupgrfs  27679  nbfiusgrfi  27723  vtxdgfusgrf  27845  usgruvtxvdb  27877  vdiscusgrb  27878  vdiscusgr  27879  fusgrn0eqdrusgr  27918  wlksnfi  28251  fusgrhashclwwlkn  28422  clwlksndivn  28429  fusgr2wsp2nb  28677  fusgreghash2wspv  28678  numclwwlk4  28729