omege0 - Mathbox for Glauco Siliprandi

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Theorem omege0 46813

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 11183	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ ℝ^*)
3		pnfxr 11190	. . 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 46783	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))
9		iccgelb 13322	. 2 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘𝐴) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐴))
10	2, 4, 8, 9	syl3anc 1374	1 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⊆ wss 3902 ∪ cuni 4864 class class class wbr 5099 dom cdm 5625 ‘cfv 6493 (class class class)co 7360 0cc0 11030 +∞cpnf 11167 ℝ^*cxr 11169 ≤ cle 11171 [,]cicc 13268 OutMeascome 46769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-addrcl 11091 ax-rnegex 11101 ax-cnre 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3401 df-v 3443 df-sbc 3742 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7363 df-oprab 7364 df-mpo 7365 df-pnf 11172 df-xr 11174 df-icc 13272 df-ome 46770

This theorem is referenced by: omess0 46814

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