Theorem fusgrusgr 29395

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 2736	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29391	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2113  ‘cfv 6492  Fincfn 8883  Vtxcvtx 29069  USGraphcusgr 29222  FinUSGraphcfusgr 29389

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-fusgr 29390

This theorem is referenced by:  fusgredgfi  29398  fusgrfisstep  29402  fusgrfupgrfs  29404  nbfiusgrfi  29448  vtxdgfusgrf  29571  usgruvtxvdb  29603  vdiscusgrb  29604  vdiscusgr  29605  fusgrn0eqdrusgr  29644  wlksnfi  29980  fusgrhashclwwlkn  30154  clwlksndivn  30161  fusgr2wsp2nb  30409  fusgreghash2wspv  30410  numclwwlk4  30461  clnbfiusgrfi  48086