Theorem fusgrusgr 27107

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 2824	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 27103	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 500	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2113  ‘cfv 6358  Fincfn 8512  Vtxcvtx 26784  USGraphcusgr 26937  FinUSGraphcfusgr 27101

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-iota 6317  df-fv 6366  df-fusgr 27102

This theorem is referenced by:  fusgredgfi  27110  fusgrfisstep  27114  fusgrfupgrfs  27116  nbfiusgrfi  27160  vtxdgfusgrf  27282  usgruvtxvdb  27314  vdiscusgrb  27315  vdiscusgr  27316  fusgrn0eqdrusgr  27355  wlksnfi  27689  fusgrhashclwwlkn  27861  clwlksndivn  27868  fusgr2wsp2nb  28116  fusgreghash2wspv  28117  numclwwlk4  28168