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| Mirrors > Home > MPE Home > Th. List > Mathboxes > omecl | Structured version Visualization version GIF version | ||
| Description: The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| omecl.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
| omecl.x | ⊢ 𝑋 = ∪ dom 𝑂 |
| omecl.ss | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
| Ref | Expression |
|---|---|
| omecl | ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | omecl.o | . . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
| 2 | omecl.x | . . 3 ⊢ 𝑋 = ∪ dom 𝑂 | |
| 3 | 1, 2 | omef 46511 | . 2 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
| 4 | omecl.ss | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
| 5 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑋 = ∪ dom 𝑂) |
| 6 | 1 | dmexd 7925 | . . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V) |
| 7 | 6 | uniexd 7762 | . . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
| 8 | 5, 7 | eqeltrd 2841 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
| 9 | 8, 4 | ssexd 5324 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
| 10 | elpwg 4603 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
| 11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
| 12 | 4, 11 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
| 13 | 3, 12 | ffvelcdmd 7105 | 1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 [,]cicc 13390 OutMeascome 46504 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ome 46505 |
| This theorem is referenced by: caragen0 46521 omexrcl 46522 caragenunidm 46523 omessre 46525 caragenuncllem 46527 caragendifcl 46529 omeunle 46531 omeiunle 46532 omeiunltfirp 46534 carageniuncllem2 46537 carageniuncl 46538 caratheodorylem1 46541 caratheodorylem2 46542 omege0 46548 |
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