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Theorem omecl 40024
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 40017 . 2 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
4 omecl.ss . . 3 (𝜑𝐴𝑋)
52a1i 11 . . . . . 6 (𝜑𝑋 = dom 𝑂)
6 dmexg 7044 . . . . . . . 8 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
71, 6syl 17 . . . . . . 7 (𝜑 → dom 𝑂 ∈ V)
8 uniexg 6908 . . . . . . 7 (dom 𝑂 ∈ V → dom 𝑂 ∈ V)
97, 8syl 17 . . . . . 6 (𝜑 dom 𝑂 ∈ V)
105, 9eqeltrd 2698 . . . . 5 (𝜑𝑋 ∈ V)
1110, 4ssexd 4765 . . . 4 (𝜑𝐴 ∈ V)
12 elpwg 4138 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1311, 12syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
144, 13mpbird 247 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
153, 14ffvelrnd 6316 1 (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402  dom cdm 5074  cfv 5847  (class class class)co 6604  0cc0 9880  +∞cpnf 10015  [,]cicc 12120  OutMeascome 40010
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
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-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ome 40011
This theorem is referenced by:  caragen0  40027  omexrcl  40028  caragenunidm  40029  omessre  40031  caragenuncllem  40033  caragendifcl  40035  omeunle  40037  omeiunle  40038  omeiunltfirp  40040  carageniuncllem2  40043  carageniuncl  40044  caratheodorylem1  40047  caratheodorylem2  40048  omege0  40054
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