Theorem omecl 47108

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 47101	. 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 7899	. . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V)
7	6	uniexd 7740	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V)
8	5, 7	eqeltrd 2869	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
9	8, 4	ssexd 5295	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
10		elpwg 4570	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
11	9, 10	syl 18	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
12	4, 11	mpbird 260	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
13	3, 12	ffvelcdmd 7081	1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4876  dom cdm 5662  ‘cfv 6537  (class class class)co 7411  0cc0 11099  +∞cpnf 11239  [,]cicc 13374  OutMeascome 47094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ome 47095

This theorem is referenced by:  caragen0  47111  omexrcl  47112  caragenunidm  47113  omessre  47115  caragenuncllem  47117  caragendifcl  47119  omeunle  47121  omeiunle  47122  omeiunltfirp  47124  carageniuncllem2  47127  carageniuncl  47128  caratheodorylem1  47131  caratheodorylem2  47132  omege0  47138