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Theorem omecl 41654
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 41647 . 2 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
4 omecl.ss . . 3 (𝜑𝐴𝑋)
52a1i 11 . . . . . 6 (𝜑𝑋 = dom 𝑂)
61dmexd 7379 . . . . . . 7 (𝜑 → dom 𝑂 ∈ V)
7 uniexg 7234 . . . . . . 7 (dom 𝑂 ∈ V → dom 𝑂 ∈ V)
86, 7syl 17 . . . . . 6 (𝜑 dom 𝑂 ∈ V)
95, 8eqeltrd 2859 . . . . 5 (𝜑𝑋 ∈ V)
109, 4ssexd 5044 . . . 4 (𝜑𝐴 ∈ V)
11 elpwg 4387 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1210, 11syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
134, 12mpbird 249 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
143, 13ffvelrnd 6626 1 (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1601  wcel 2107  Vcvv 3398  wss 3792  𝒫 cpw 4379   cuni 4673  dom cdm 5357  cfv 6137  (class class class)co 6924  0cc0 10274  +∞cpnf 10410  [,]cicc 12494  OutMeascome 41640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ome 41641
This theorem is referenced by:  caragen0  41657  omexrcl  41658  caragenunidm  41659  omessre  41661  caragenuncllem  41663  caragendifcl  41665  omeunle  41667  omeiunle  41668  omeiunltfirp  41670  carageniuncllem2  41673  carageniuncl  41674  caratheodorylem1  41677  caratheodorylem2  41678  omege0  41684
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