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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 3624 | . . 3 ⊢ 𝒫 𝑂 ⊆ 𝒫 𝑂 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂) |
3 | pwidg 4173 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂) | |
4 | difss 3737 | . . . . . 6 ⊢ (𝑂 ∖ 𝑥) ⊆ 𝑂 | |
5 | elpw2g 4827 | . . . . . 6 ⊢ (𝑂 ∈ 𝑉 → ((𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ↔ (𝑂 ∖ 𝑥) ⊆ 𝑂)) | |
6 | 4, 5 | mpbiri 248 | . . . . 5 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂) |
7 | 6 | a1d 25 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝑂 → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)) |
8 | 7 | ralrimiv 2965 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂) |
9 | sspwuni 4611 | . . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂) | |
10 | vuniex 6954 | . . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V | |
11 | 10 | elpw 4164 | . . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂) |
12 | 9, 11 | bitr4i 267 | . . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ∈ 𝒫 𝑂) |
13 | 12 | biimpi 206 | . . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝑂 → ∪ 𝑥 ∈ 𝒫 𝑂) |
14 | 13 | a1d 25 | . . . . 5 ⊢ (𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)) |
15 | elpwi 4168 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → 𝑥 ⊆ 𝒫 𝑂) | |
16 | 15 | imim1i 63 | . . . . 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 2965 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)) |
19 | 3, 8, 18 | 3jca 1242 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))) |
20 | pwexg 4850 | . . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V) | |
21 | issiga 30174 | . . 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 957 | 1 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 class class class wbr 4653 ‘cfv 5888 ωcom 7065 ≼ cdom 7953 sigAlgebracsiga 30170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-siga 30171 |
This theorem is referenced by: sigagenval 30203 dmsigagen 30207 ldsysgenld 30223 pwcntmeas 30290 ddemeas 30299 mbfmcnt 30330 |
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