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Theorem omecl 44091
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 44084 . 2 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
4 omecl.ss . . 3 (𝜑𝐴𝑋)
52a1i 11 . . . . . 6 (𝜑𝑋 = dom 𝑂)
61dmexd 7784 . . . . . . 7 (𝜑 → dom 𝑂 ∈ V)
76uniexd 7627 . . . . . 6 (𝜑 dom 𝑂 ∈ V)
85, 7eqeltrd 2837 . . . . 5 (𝜑𝑋 ∈ V)
98, 4ssexd 5257 . . . 4 (𝜑𝐴 ∈ V)
10 elpwg 4542 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
119, 10syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
124, 11mpbird 257 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
133, 12ffvelcdmd 6994 1 (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2104  Vcvv 3437  wss 3892  𝒫 cpw 4539   cuni 4844  dom cdm 5600  cfv 6458  (class class class)co 7307  0cc0 10917  +∞cpnf 11052  [,]cicc 13128  OutMeascome 44077
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466  df-ome 44078
This theorem is referenced by:  caragen0  44094  omexrcl  44095  caragenunidm  44096  omessre  44098  caragenuncllem  44100  caragendifcl  44102  omeunle  44104  omeiunle  44105  omeiunltfirp  44107  carageniuncllem2  44110  carageniuncl  44111  caratheodorylem1  44114  caratheodorylem2  44115  omege0  44121
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