omege0 - Mathbox for Glauco Siliprandi

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Theorem omege0 42822

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 10690	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ ℝ^*)
3		pnfxr 10697	. . 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 42792	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))
9		iccgelb 12796	. 2 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘𝐴) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐴))
10	2, 4, 8, 9	syl3anc 1367	1 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 ℝ^*cxr 10676 ≤ cle 10678 [,]cicc 12744 OutMeascome 42778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-addrcl 10600 ax-rnegex 10610 ax-cnre 10612

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-pnf 10679 df-xr 10681 df-icc 12748 df-ome 42779

This theorem is referenced by: omess0 42823

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