Theorem fusgrusgr 29405

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 29401	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 496	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6492  Fincfn 8886  Vtxcvtx 29079  USGraphcusgr 29232  FinUSGraphcfusgr 29399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-fusgr 29400

This theorem is referenced by:  fusgredgfi  29408  fusgrfisstep  29412  fusgrfupgrfs  29414  nbfiusgrfi  29458  vtxdgfusgrf  29581  usgruvtxvdb  29613  vdiscusgrb  29614  vdiscusgr  29615  fusgrn0eqdrusgr  29654  wlksnfi  29990  fusgrhashclwwlkn  30164  clwlksndivn  30171  fusgr2wsp2nb  30419  fusgreghash2wspv  30420  numclwwlk4  30471  clnbfiusgrfi  48332