omexrcl - Mathbox for Glauco Siliprandi

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Theorem omexrcl 41648

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

**Hypotheses**
Ref	Expression
omexrcl.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
omexrcl.x	⊢ 𝑋 = ∪ dom 𝑂
omexrcl.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)

**Assertion**
Ref	Expression
omexrcl	⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ^*)

**Proof of Theorem omexrcl**
Step	Hyp	Ref	Expression
1		iccssxr 12568	. 2 ⊢ (0[,]+∞) ⊆ ℝ^*
2		omexrcl.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
3		omexrcl.x	. . 3 ⊢ 𝑋 = ∪ dom 𝑂
4		omexrcl.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
5	2, 3, 4	omecl 41644	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))
6	1, 5	sseldi 3819	1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 ∪ cuni 4671 dom cdm 5355 ‘cfv 6135 (class class class)co 6922 0cc0 10272 +∞cpnf 10408 ℝ^*cxr 10410 [,]cicc 12490 OutMeascome 41630

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-xr 10415 df-icc 12494 df-ome 41631

This theorem is referenced by: omessre 41651 caragenuncllem 41653 omeiunltfirp 41660 caratheodorylem1 41667 caratheodorylem2 41668 caragenel2d 41673 omess0 41675 caragencmpl 41676

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