Theorem fusgrusgr 29391

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 2736	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29387	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 496	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6498  Fincfn 8893  Vtxcvtx 29065  USGraphcusgr 29218  FinUSGraphcfusgr 29385

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-fusgr 29386

This theorem is referenced by:  fusgredgfi  29394  fusgrfisstep  29398  fusgrfupgrfs  29400  nbfiusgrfi  29444  vtxdgfusgrf  29566  usgruvtxvdb  29598  vdiscusgrb  29599  vdiscusgr  29600  fusgrn0eqdrusgr  29639  wlksnfi  29975  fusgrhashclwwlkn  30149  clwlksndivn  30156  fusgr2wsp2nb  30404  fusgreghash2wspv  30405  numclwwlk4  30456  clnbfiusgrfi  48320