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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsigagen2 | Structured version Visualization version GIF version |
Description: Any countable union of elements of a set is also in the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.) |
Ref | Expression |
---|---|
elsigagen2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1127 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐴 ∈ 𝑉) | |
2 | 1 | sgsiga 30803 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra) |
3 | sssigagen 30806 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴)) | |
4 | sspwb 5149 | . . . . 5 ⊢ (𝐴 ⊆ (sigaGen‘𝐴) ↔ 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴)) | |
5 | 4 | biimpi 208 | . . . 4 ⊢ (𝐴 ⊆ (sigaGen‘𝐴) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴)) |
6 | 1, 3, 5 | 3syl 18 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴)) |
7 | simp2 1128 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ⊆ 𝐴) | |
8 | simp3 1129 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω) | |
9 | ctex 8256 | . . . . 5 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
10 | elpwg 4387 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴)) | |
11 | 8, 9, 10 | 3syl 18 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴)) |
12 | 7, 11 | mpbird 249 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ∈ 𝒫 𝐴) |
13 | 6, 12 | sseldd 3822 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴)) |
14 | sigaclcu 30778 | . 2 ⊢ (((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴)) | |
15 | 2, 13, 8, 14 | syl3anc 1439 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1071 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 𝒫 cpw 4379 ∪ cuni 4671 class class class wbr 4886 ran crn 5356 ‘cfv 6135 ωcom 7343 ≼ cdom 8239 sigAlgebracsiga 30768 sigaGencsigagen 30799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-int 4711 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-fv 6143 df-dom 8243 df-siga 30769 df-sigagen 30800 |
This theorem is referenced by: sxbrsigalem1 30945 |
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