Theorem fusgrusgr 29339

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 2737	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29335	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6561  Fincfn 8985  Vtxcvtx 29013  USGraphcusgr 29166  FinUSGraphcfusgr 29333

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-fusgr 29334

This theorem is referenced by:  fusgredgfi  29342  fusgrfisstep  29346  fusgrfupgrfs  29348  nbfiusgrfi  29392  vtxdgfusgrf  29515  usgruvtxvdb  29547  vdiscusgrb  29548  vdiscusgr  29549  fusgrn0eqdrusgr  29588  wlksnfi  29927  fusgrhashclwwlkn  30098  clwlksndivn  30105  fusgr2wsp2nb  30353  fusgreghash2wspv  30354  numclwwlk4  30405  clnbfiusgrfi  47830