Theorem fusgrusgr 29354

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 2735	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29350	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6563  Fincfn 8984  Vtxcvtx 29028  USGraphcusgr 29181  FinUSGraphcfusgr 29348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-fusgr 29349

This theorem is referenced by:  fusgredgfi  29357  fusgrfisstep  29361  fusgrfupgrfs  29363  nbfiusgrfi  29407  vtxdgfusgrf  29530  usgruvtxvdb  29562  vdiscusgrb  29563  vdiscusgr  29564  fusgrn0eqdrusgr  29603  wlksnfi  29937  fusgrhashclwwlkn  30108  clwlksndivn  30115  fusgr2wsp2nb  30363  fusgreghash2wspv  30364  numclwwlk4  30415  clnbfiusgrfi  47768