Theorem fusgrusgr 29357

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 2740	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29353	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6573  Fincfn 9003  Vtxcvtx 29031  USGraphcusgr 29184  FinUSGraphcfusgr 29351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-fusgr 29352

This theorem is referenced by:  fusgredgfi  29360  fusgrfisstep  29364  fusgrfupgrfs  29366  nbfiusgrfi  29410  vtxdgfusgrf  29533  usgruvtxvdb  29565  vdiscusgrb  29566  vdiscusgr  29567  fusgrn0eqdrusgr  29606  wlksnfi  29940  fusgrhashclwwlkn  30111  clwlksndivn  30118  fusgr2wsp2nb  30366  fusgreghash2wspv  30367  numclwwlk4  30418  clnbfiusgrfi  47716