Theorem omecl 46518

Description: The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
omecl.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
omecl.x	⊢ 𝑋 = ∪ dom 𝑂
omecl.ss	⊢ (𝜑 → 𝐴 ⊆ 𝑋)

Assertion

Ref	Expression
omecl	⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))

Proof of Theorem omecl

Step	Hyp	Ref	Expression
1		omecl.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		omecl.x	. . 3 ⊢ 𝑋 = ∪ dom 𝑂
3	1, 2	omef 46511	. 2 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))
4		omecl.ss	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
5	2	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑋 = ∪ dom 𝑂)
6	1	dmexd 7925	. . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V)
7	6	uniexd 7762	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V)
8	5, 7	eqeltrd 2841	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
9	8, 4	ssexd 5324	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
10		elpwg 4603	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
11	9, 10	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
12	4, 11	mpbird 257	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
13	3, 12	ffvelcdmd 7105	1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  dom cdm 5685  ‘cfv 6561  (class class class)co 7431  0cc0 11155  +∞cpnf 11292  [,]cicc 13390  OutMeascome 46504

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ome 46505

This theorem is referenced by:  caragen0  46521  omexrcl  46522  caragenunidm  46523  omessre  46525  caragenuncllem  46527  caragendifcl  46529  omeunle  46531  omeiunle  46532  omeiunltfirp  46534  carageniuncllem2  46537  carageniuncl  46538  caratheodorylem1  46541  caratheodorylem2  46542  omege0  46548