Theorem fusgrusgr 29936

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 2763	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29932	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 502	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ‘cfv 6561  Fincfn 9004  Vtxcvtx 29607  USGraphcusgr 29763  FinUSGraphcfusgr 29930

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-fusgr 29931

This theorem is used by:  fusgredgfi  29939  fusgrfisstep  29943  fusgrfupgrfs  29945  nbfiusgrfi  29989  vtxdgfusgrf  30111  usgruvtxvdb  30143  vdiscusgrb  30144  vdiscusgr  30145  fusgrn0eqdrusgr  30184  wlksnfi  30529  fusgrhashclwwlkn  30703  clwlksndivn  30710  fusgr2wsp2nb  30968  fusgreghash2wspv  30969  numclwwlk4  31020  clnbfiusgrfi  49157