Theorem fusgrusgr 29285

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 2729	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29281	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2109  ‘cfv 6486  Fincfn 8879  Vtxcvtx 28959  USGraphcusgr 29112  FinUSGraphcfusgr 29279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-fusgr 29280

This theorem is referenced by:  fusgredgfi  29288  fusgrfisstep  29292  fusgrfupgrfs  29294  nbfiusgrfi  29338  vtxdgfusgrf  29461  usgruvtxvdb  29493  vdiscusgrb  29494  vdiscusgr  29495  fusgrn0eqdrusgr  29534  wlksnfi  29870  fusgrhashclwwlkn  30041  clwlksndivn  30048  fusgr2wsp2nb  30296  fusgreghash2wspv  30297  numclwwlk4  30348  clnbfiusgrfi  47829