omege0 - Mathbox for Glauco Siliprandi

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Theorem omege0 46553

Description: If the outer measure of a set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
omege0.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
omege0.x	⊢ 𝑋 = ∪ dom 𝑂
omege0.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)

**Assertion**
Ref	Expression
omege0	⊢ (𝜑 → 0 ≤ (𝑂‘𝐴))

**Proof of Theorem omege0**
Step	Hyp	Ref	Expression
1		0xr 11309	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ ℝ^*)
3		pnfxr 11316	. . 3 ⊢ +∞ ∈ ℝ^*
4	3	a1i 11	. 2 ⊢ (𝜑 → +∞ ∈ ℝ^*)
5		omege0.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
6		omege0.x	. . 3 ⊢ 𝑋 = ∪ dom 𝑂
7		omege0.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
8	5, 6, 7	omecl 46523	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))
9		iccgelb 13444	. 2 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘𝐴) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐴))
10	2, 4, 8, 9	syl3anc 1372	1 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ∪ cuni 4906 class class class wbr 5142 dom cdm 5684 ‘cfv 6560 (class class class)co 7432 0cc0 11156 +∞cpnf 11293 ℝ^*cxr 11295 ≤ cle 11297 [,]cicc 13391 OutMeascome 46509

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-addrcl 11217 ax-rnegex 11227 ax-cnre 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-pnf 11298 df-xr 11300 df-icc 13395 df-ome 46510

This theorem is referenced by: omess0 46554

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