Theorem fusgrusgr 28847

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 2731	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 28843	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2105  ‘cfv 6543  Fincfn 8943  Vtxcvtx 28524  USGraphcusgr 28677  FinUSGraphcfusgr 28841

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-fusgr 28842

This theorem is referenced by:  fusgredgfi  28850  fusgrfisstep  28854  fusgrfupgrfs  28856  nbfiusgrfi  28900  vtxdgfusgrf  29022  usgruvtxvdb  29054  vdiscusgrb  29055  vdiscusgr  29056  fusgrn0eqdrusgr  29095  wlksnfi  29429  fusgrhashclwwlkn  29600  clwlksndivn  29607  fusgr2wsp2nb  29855  fusgreghash2wspv  29856  numclwwlk4  29907