Theorem fusgrusgr 29581

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 2765	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29577	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 501	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2145  ‘cfv 6525  Fincfn 8931  Vtxcvtx 29255  USGraphcusgr 29408  FinUSGraphcfusgr 29575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-fusgr 29576

This theorem is referenced by:  fusgredgfi  29584  fusgrfisstep  29588  fusgrfupgrfs  29590  nbfiusgrfi  29634  vtxdgfusgrf  29756  usgruvtxvdb  29788  vdiscusgrb  29789  vdiscusgr  29790  fusgrn0eqdrusgr  29829  wlksnfi  30165  fusgrhashclwwlkn  30339  clwlksndivn  30346  fusgr2wsp2nb  30594  fusgreghash2wspv  30595  numclwwlk4  30646  clnbfiusgrfi  48464