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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salunicl | Structured version Visualization version GIF version |
Description: SAlg sigma-algebra is closed under countable union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
salunicl.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salunicl.t | ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆) |
salunicl.tct | ⊢ (𝜑 → 𝑇 ≼ ω) |
Ref | Expression |
---|---|
salunicl | ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salunicl.tct | . 2 ⊢ (𝜑 → 𝑇 ≼ ω) | |
2 | breq1 5108 | . . . 4 ⊢ (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω)) | |
3 | unieq 4876 | . . . . 5 ⊢ (𝑦 = 𝑇 → ∪ 𝑦 = ∪ 𝑇) | |
4 | 3 | eleq1d 2822 | . . . 4 ⊢ (𝑦 = 𝑇 → (∪ 𝑦 ∈ 𝑆 ↔ ∪ 𝑇 ∈ 𝑆)) |
5 | 2, 4 | imbi12d 344 | . . 3 ⊢ (𝑦 = 𝑇 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆) ↔ (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆))) |
6 | salunicl.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
7 | issal 44526 | . . . . . 6 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) |
9 | 6, 8 | mpbid 231 | . . . 4 ⊢ (𝜑 → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) |
10 | 9 | simp3d 1144 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
11 | salunicl.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆) | |
12 | 5, 10, 11 | rspcdva 3582 | . 2 ⊢ (𝜑 → (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆)) |
13 | 1, 12 | mpd 15 | 1 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∖ cdif 3907 ∅c0 4282 𝒫 cpw 4560 ∪ cuni 4865 class class class wbr 5105 ωcom 7801 ≼ cdom 8880 SAlgcsalg 44520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-salg 44521 |
This theorem is referenced by: saliunclf 44534 intsal 44542 smfpimbor1lem1 45010 |
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