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Theorem omecl 41511
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 41504 . 2 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
4 omecl.ss . . 3 (𝜑𝐴𝑋)
52a1i 11 . . . . . 6 (𝜑𝑋 = dom 𝑂)
61dmexd 7360 . . . . . . 7 (𝜑 → dom 𝑂 ∈ V)
7 uniexg 7215 . . . . . . 7 (dom 𝑂 ∈ V → dom 𝑂 ∈ V)
86, 7syl 17 . . . . . 6 (𝜑 dom 𝑂 ∈ V)
95, 8eqeltrd 2906 . . . . 5 (𝜑𝑋 ∈ V)
109, 4ssexd 5030 . . . 4 (𝜑𝐴 ∈ V)
11 elpwg 4386 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1210, 11syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
134, 12mpbird 249 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
143, 13ffvelrnd 6609 1 (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1658  wcel 2166  Vcvv 3414  wss 3798  𝒫 cpw 4378   cuni 4658  dom cdm 5342  cfv 6123  (class class class)co 6905  0cc0 10252  +∞cpnf 10388  [,]cicc 12466  OutMeascome 41497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ome 41498
This theorem is referenced by:  caragen0  41514  omexrcl  41515  caragenunidm  41516  omessre  41518  caragenuncllem  41520  caragendifcl  41522  omeunle  41524  omeiunle  41525  omeiunltfirp  41527  carageniuncllem2  41530  carageniuncl  41531  caratheodorylem1  41534  caratheodorylem2  41535  omege0  41541
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