Theorem fusgrusgr 29709

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 2766	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29705	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 502	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  ‘cfv 6543  Fincfn 8952  Vtxcvtx 29383  USGraphcusgr 29536  FinUSGraphcfusgr 29703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-fusgr 29704

This theorem is used by:  fusgredgfi  29712  fusgrfisstep  29716  fusgrfupgrfs  29718  nbfiusgrfi  29762  vtxdgfusgrf  29884  usgruvtxvdb  29916  vdiscusgrb  29917  vdiscusgr  29918  fusgrn0eqdrusgr  29957  wlksnfi  30293  fusgrhashclwwlkn  30467  clwlksndivn  30474  fusgr2wsp2nb  30722  fusgreghash2wspv  30723  numclwwlk4  30774  clnbfiusgrfi  48650