Theorem fusgrusgr 29302

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 2733	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29298	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2113  ‘cfv 6486  Fincfn 8875  Vtxcvtx 28976  USGraphcusgr 29129  FinUSGraphcfusgr 29296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-fusgr 29297

This theorem is referenced by:  fusgredgfi  29305  fusgrfisstep  29309  fusgrfupgrfs  29311  nbfiusgrfi  29355  vtxdgfusgrf  29478  usgruvtxvdb  29510  vdiscusgrb  29511  vdiscusgr  29512  fusgrn0eqdrusgr  29551  wlksnfi  29887  fusgrhashclwwlkn  30061  clwlksndivn  30068  fusgr2wsp2nb  30316  fusgreghash2wspv  30317  numclwwlk4  30368  clnbfiusgrfi  47968