Theorem fusgrusgr 29249

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 2729	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29245	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2109  ‘cfv 6511  Fincfn 8918  Vtxcvtx 28923  USGraphcusgr 29076  FinUSGraphcfusgr 29243

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-fusgr 29244

This theorem is referenced by:  fusgredgfi  29252  fusgrfisstep  29256  fusgrfupgrfs  29258  nbfiusgrfi  29302  vtxdgfusgrf  29425  usgruvtxvdb  29457  vdiscusgrb  29458  vdiscusgr  29459  fusgrn0eqdrusgr  29498  wlksnfi  29837  fusgrhashclwwlkn  30008  clwlksndivn  30015  fusgr2wsp2nb  30263  fusgreghash2wspv  30264  numclwwlk4  30315  clnbfiusgrfi  47844