Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pwsiga | Structured version Visualization version GIF version |
Description: Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.) |
Ref | Expression |
---|---|
pwsiga | ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3587 | . . 3 ⊢ 𝒫 𝑂 ⊆ 𝒫 𝑂 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂) |
3 | pwidg 4121 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂) | |
4 | difss 3699 | . . . . . 6 ⊢ (𝑂 ∖ 𝑥) ⊆ 𝑂 | |
5 | elpw2g 4754 | . . . . . 6 ⊢ (𝑂 ∈ 𝑉 → ((𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ↔ (𝑂 ∖ 𝑥) ⊆ 𝑂)) | |
6 | 4, 5 | mpbiri 247 | . . . . 5 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂) |
7 | 6 | a1d 25 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝑂 → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)) |
8 | 7 | ralrimiv 2948 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂) |
9 | sspwuni 4547 | . . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂) | |
10 | vuniex 6852 | . . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V | |
11 | 10 | elpw 4114 | . . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂) |
12 | 9, 11 | bitr4i 266 | . . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ∈ 𝒫 𝑂) |
13 | 12 | biimpi 205 | . . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝑂 → ∪ 𝑥 ∈ 𝒫 𝑂) |
14 | 13 | a1d 25 | . . . . 5 ⊢ (𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)) |
15 | elpwi 4117 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → 𝑥 ⊆ 𝒫 𝑂) | |
16 | 15 | imim1i 61 | . . . . 5 ⊢ ((𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)) → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))) |
17 | 14, 16 | mp1i 13 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))) |
18 | 17 | ralrimiv 2948 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)) |
19 | 3, 8, 18 | 3jca 1235 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))) |
20 | pwexg 4776 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V) | |
21 | issiga 29501 | . . 3 ⊢ (𝒫 𝑂 ∈ V → (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) ↔ (𝒫 𝑂 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))))) | |
22 | 20, 21 | syl 17 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) ↔ (𝒫 𝑂 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))))) |
23 | 2, 19, 22 | mpbir2and 959 | 1 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 class class class wbr 4583 ‘cfv 5804 ωcom 6957 ≼ cdom 7839 sigAlgebracsiga 29497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-siga 29498 |
This theorem is referenced by: sigagenval 29530 dmsigagen 29534 ldsysgenld 29550 pwcntmeas 29617 ddemeas 29626 mbfmcnt 29657 |
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