Theorem fusgrusgr 29469

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 2761	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29465	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 500	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2141  ‘cfv 6517  Fincfn 8923  Vtxcvtx 29143  USGraphcusgr 29296  FinUSGraphcfusgr 29463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-fusgr 29464

This theorem is referenced by:  fusgredgfi  29472  fusgrfisstep  29476  fusgrfupgrfs  29478  nbfiusgrfi  29522  vtxdgfusgrf  29644  usgruvtxvdb  29676  vdiscusgrb  29677  vdiscusgr  29678  fusgrn0eqdrusgr  29717  wlksnfi  30053  fusgrhashclwwlkn  30227  clwlksndivn  30234  fusgr2wsp2nb  30482  fusgreghash2wspv  30483  numclwwlk4  30534  clnbfiusgrfi  48430