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Theorem omecl 46518
Description: The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
omecl.o (𝜑𝑂 ∈ OutMeas)
omecl.x 𝑋 = dom 𝑂
omecl.ss (𝜑𝐴𝑋)
Assertion
Ref Expression
omecl (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))

Proof of Theorem omecl
StepHypRef Expression
1 omecl.o . . 3 (𝜑𝑂 ∈ OutMeas)
2 omecl.x . . 3 𝑋 = dom 𝑂
31, 2omef 46511 . 2 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
4 omecl.ss . . 3 (𝜑𝐴𝑋)
52a1i 11 . . . . . 6 (𝜑𝑋 = dom 𝑂)
61dmexd 7925 . . . . . . 7 (𝜑 → dom 𝑂 ∈ V)
76uniexd 7762 . . . . . 6 (𝜑 dom 𝑂 ∈ V)
85, 7eqeltrd 2841 . . . . 5 (𝜑𝑋 ∈ V)
98, 4ssexd 5324 . . . 4 (𝜑𝐴 ∈ V)
10 elpwg 4603 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
119, 10syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
124, 11mpbird 257 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
133, 12ffvelcdmd 7105 1 (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  𝒫 cpw 4600   cuni 4907  dom cdm 5685  cfv 6561  (class class class)co 7431  0cc0 11155  +∞cpnf 11292  [,]cicc 13390  OutMeascome 46504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ome 46505
This theorem is referenced by:  caragen0  46521  omexrcl  46522  caragenunidm  46523  omessre  46525  caragenuncllem  46527  caragendifcl  46529  omeunle  46531  omeiunle  46532  omeiunltfirp  46534  carageniuncllem2  46537  carageniuncl  46538  caratheodorylem1  46541  caratheodorylem2  46542  omege0  46548
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