omege0 - Mathbox for Glauco Siliprandi

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Theorem omege0 46531

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 11221	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ ℝ^*)
3		pnfxr 11228	. . 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 46501	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))
9		iccgelb 13363	. 2 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘𝐴) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐴))
10	2, 4, 8, 9	syl3anc 1373	1 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ⊆ wss 3914 ∪ cuni 4871 class class class wbr 5107 dom cdm 5638 ‘cfv 6511 (class class class)co 7387 0cc0 11068 +∞cpnf 11205 ℝ^*cxr 11207 ≤ cle 11209 [,]cicc 13309 OutMeascome 46487

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-addrcl 11129 ax-rnegex 11139 ax-cnre 11141

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-pnf 11210 df-xr 11212 df-icc 13313 df-ome 46488

This theorem is referenced by: omess0 46532

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