![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 46153 | . 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 7908 | . . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V) |
7 | 6 | uniexd 7745 | . . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
8 | 5, 7 | eqeltrd 2826 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
9 | 8, 4 | ssexd 5321 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
10 | elpwg 4600 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
12 | 4, 11 | mpbird 256 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
13 | 3, 12 | ffvelcdmd 7091 | 1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3946 𝒫 cpw 4597 ∪ cuni 4905 dom cdm 5674 ‘cfv 6546 (class class class)co 7416 0cc0 11149 +∞cpnf 11286 [,]cicc 13375 OutMeascome 46146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fv 6554 df-ome 46147 |
This theorem is referenced by: caragen0 46163 omexrcl 46164 caragenunidm 46165 omessre 46167 caragenuncllem 46169 caragendifcl 46171 omeunle 46173 omeiunle 46174 omeiunltfirp 46176 carageniuncllem2 46179 carageniuncl 46180 caratheodorylem1 46183 caratheodorylem2 46184 omege0 46190 |
Copyright terms: Public domain | W3C validator |