Theorem fusgrusgr 28846

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 2730	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 28842	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 496	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2104  ‘cfv 6542  Fincfn 8941  Vtxcvtx 28523  USGraphcusgr 28676  FinUSGraphcfusgr 28840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  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-iota 6494  df-fv 6550  df-fusgr 28841

This theorem is referenced by:  fusgredgfi  28849  fusgrfisstep  28853  fusgrfupgrfs  28855  nbfiusgrfi  28899  vtxdgfusgrf  29021  usgruvtxvdb  29053  vdiscusgrb  29054  vdiscusgr  29055  fusgrn0eqdrusgr  29094  wlksnfi  29428  fusgrhashclwwlkn  29599  clwlksndivn  29606  fusgr2wsp2nb  29854  fusgreghash2wspv  29855  numclwwlk4  29906