Theorem fusgrusgr 26669

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 2777	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 26665	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 493	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6135  Fincfn 8241  Vtxcvtx 26344  USGraphcusgr 26498  FinUSGraphcfusgr 26663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-fusgr 26664

This theorem is referenced by:  fusgredgfi  26672  fusgrfisstep  26676  fusgrfupgrfs  26678  nbfiusgrfi  26723  vtxdgfusgrf  26845  usgruvtxvdb  26877  vdiscusgrb  26878  vdiscusgr  26879  fusgrn0eqdrusgr  26918  wlksnfi  27280  fusgrhashclwwlkn  27477  clwlksndivn  27488  fusgr2wsp2nb  27742  fusgreghash2wspv  27743  numclwwlk4  27818