Theorem fusgrusgr 29790

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 2762	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29786	. 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 2145  ‘cfv 6537  Fincfn 8956  Vtxcvtx 29461  USGraphcusgr 29617  FinUSGraphcfusgr 29784

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 2147  ax-9 2155  ax-ext 2734

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 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-fusgr 29785

This theorem is used by:  fusgredgfi  29793  fusgrfisstep  29797  fusgrfupgrfs  29799  nbfiusgrfi  29843  vtxdgfusgrf  29965  usgruvtxvdb  29997  vdiscusgrb  29998  vdiscusgr  29999  fusgrn0eqdrusgr  30038  wlksnfi  30383  fusgrhashclwwlkn  30557  clwlksndivn  30564  fusgr2wsp2nb  30822  fusgreghash2wspv  30823  numclwwlk4  30874  clnbfiusgrfi  48768