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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 | ssidd 4003 | . 2 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂) | |
2 | pwidg 4627 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂) | |
3 | difss 4131 | . . . . . 6 ⊢ (𝑂 ∖ 𝑥) ⊆ 𝑂 | |
4 | elpw2g 5351 | . . . . . 6 ⊢ (𝑂 ∈ 𝑉 → ((𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ↔ (𝑂 ∖ 𝑥) ⊆ 𝑂)) | |
5 | 3, 4 | mpbiri 257 | . . . . 5 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂) |
6 | 5 | a1d 25 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝑂 → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)) |
7 | 6 | ralrimiv 3135 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂) |
8 | sspwuni 5108 | . . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂) | |
9 | vuniex 7750 | . . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V | |
10 | 9 | elpw 4611 | . . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂) |
11 | 8, 10 | bitr4i 277 | . . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ∈ 𝒫 𝑂) |
12 | 11 | biimpi 215 | . . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝑂 → ∪ 𝑥 ∈ 𝒫 𝑂) |
13 | 12 | a1d 25 | . . . . 5 ⊢ (𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)) |
14 | elpwi 4614 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → 𝑥 ⊆ 𝒫 𝑂) | |
15 | 14 | imim1i 63 | . . . . 5 ⊢ ((𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)) → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))) |
16 | 13, 15 | mp1i 13 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))) |
17 | 16 | ralrimiv 3135 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)) |
18 | 2, 7, 17 | 3jca 1125 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))) |
19 | pwexg 5382 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V) | |
20 | issiga 33945 | . . 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))))) | |
21 | 19, 20 | 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))))) |
22 | 1, 18, 21 | mpbir2and 711 | 1 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ∖ cdif 3944 ⊆ wss 3947 𝒫 cpw 4607 ∪ cuni 4913 class class class wbr 5153 ‘cfv 6554 ωcom 7876 ≼ cdom 8972 sigAlgebracsiga 33941 |
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-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-siga 33942 |
This theorem is referenced by: sigagenval 33973 dmsigagen 33977 ldsysgenld 33993 pwcntmeas 34060 ddemeas 34069 mbfmcnt 34102 |
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