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Theorem fusgrusgr 26136
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
StepHypRef Expression
1 eqid 2621 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
21isfusgr 26132 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
32simplbi 476 1 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cfv 5857  Fincfn 7915  Vtxcvtx 25808   USGraph cusgr 25971   FinUSGraph cfusgr 26130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-fusgr 26131
This theorem is referenced by:  fusgredgfi  26139  fusgrfisstep  26143  nbfiusgrfi  26198  vtxdgfusgrf  26313  usgruvtxvdb  26345  vdiscusgrb  26346  vdiscusgr  26347  fusgrn0eqdrusgr  26370  wlksnfi  26705  fusgrhashclwwlkn  26856  clwlksfclwwlk  26862  clwlksfoclwwlk  26863  clwlksf1clwwlk  26869  fusgr2wsp2nb  27090  fusgreghash2wspv  27091  numclwwlk4  27132
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