Theorem fusgrusgr 27106

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 2823	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 27102	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 500	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6357  Fincfn 8511  Vtxcvtx 26783  USGraphcusgr 26936  FinUSGraphcfusgr 27100

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-fusgr 27101

This theorem is referenced by:  fusgredgfi  27109  fusgrfisstep  27113  fusgrfupgrfs  27115  nbfiusgrfi  27159  vtxdgfusgrf  27281  usgruvtxvdb  27313  vdiscusgrb  27314  vdiscusgr  27315  fusgrn0eqdrusgr  27354  wlksnfi  27688  fusgrhashclwwlkn  27860  clwlksndivn  27867  fusgr2wsp2nb  28115  fusgreghash2wspv  28116  numclwwlk4  28167