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Theorem omecl 46160
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 46153 . 2 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
4 omecl.ss . . 3 (𝜑𝐴𝑋)
52a1i 11 . . . . . 6 (𝜑𝑋 = dom 𝑂)
61dmexd 7908 . . . . . . 7 (𝜑 → dom 𝑂 ∈ V)
76uniexd 7745 . . . . . 6 (𝜑 dom 𝑂 ∈ V)
85, 7eqeltrd 2826 . . . . 5 (𝜑𝑋 ∈ V)
98, 4ssexd 5321 . . . 4 (𝜑𝐴 ∈ V)
10 elpwg 4600 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
119, 10syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
124, 11mpbird 256 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
133, 12ffvelcdmd 7091 1 (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  Vcvv 3462  wss 3946  𝒫 cpw 4597   cuni 4905  dom cdm 5674  cfv 6546  (class class class)co 7416  0cc0 11149  +∞cpnf 11286  [,]cicc 13375  OutMeascome 46146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ome 46147
This theorem is referenced by:  caragen0  46163  omexrcl  46164  caragenunidm  46165  omessre  46167  caragenuncllem  46169  caragendifcl  46171  omeunle  46173  omeiunle  46174  omeiunltfirp  46176  carageniuncllem2  46179  carageniuncl  46180  caratheodorylem1  46183  caratheodorylem2  46184  omege0  46190
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