omege0 - Mathbox for Glauco Siliprandi

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Theorem omege0 44071

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 11022	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ ℝ^*)
3		pnfxr 11029	. . 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 44041	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))
9		iccgelb 13135	. 2 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘𝐴) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐴))
10	2, 4, 8, 9	syl3anc 1370	1 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ∪ cuni 4839 class class class wbr 5074 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 0cc0 10871 +∞cpnf 11006 ℝ^*cxr 11008 ≤ cle 11010 [,]cicc 13082 OutMeascome 44027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-addrcl 10932 ax-rnegex 10942 ax-cnre 10944

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-pnf 11011 df-xr 11013 df-icc 13086 df-ome 44028

This theorem is referenced by: omess0 44072

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