Theorem omecl 46459

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 46452	. 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 7926	. . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V)
7	6	uniexd 7761	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V)
8	5, 7	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
9	8, 4	ssexd 5330	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
10		elpwg 4608	. . . 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 1537   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912  dom cdm 5689  ‘cfv 6563  (class class class)co 7431  0cc0 11153  +∞cpnf 11290  [,]cicc 13387  OutMeascome 46445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ome 46446

This theorem is referenced by:  caragen0  46462  omexrcl  46463  caragenunidm  46464  omessre  46466  caragenuncllem  46468  caragendifcl  46470  omeunle  46472  omeiunle  46473  omeiunltfirp  46475  carageniuncllem2  46478  carageniuncl  46479  caratheodorylem1  46482  caratheodorylem2  46483  omege0  46489