Theorem fusgrusgr 29653

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 2763	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29649	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3	2	simplbi 501	1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2143  ‘cfv 6538  Fincfn 8944  Vtxcvtx 29327  USGraphcusgr 29480  FinUSGraphcfusgr 29647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-fusgr 29648

This theorem is referenced by:  fusgredgfi  29656  fusgrfisstep  29660  fusgrfupgrfs  29662  nbfiusgrfi  29706  vtxdgfusgrf  29828  usgruvtxvdb  29860  vdiscusgrb  29861  vdiscusgr  29862  fusgrn0eqdrusgr  29901  wlksnfi  30237  fusgrhashclwwlkn  30411  clwlksndivn  30418  fusgr2wsp2nb  30666  fusgreghash2wspv  30667  numclwwlk4  30718  clnbfiusgrfi  48592