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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 5146 | . . . 4 ⊢ (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω)) | |
| 3 | unieq 4918 | . . . . 5 ⊢ (𝑦 = 𝑇 → ∪ 𝑦 = ∪ 𝑇) | |
| 4 | 3 | eleq1d 2826 | . . . 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 46329 | . . . . . 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 232 | . . . 4 ⊢ (𝜑 → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) | 
| 10 | 9 | simp3d 1145 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) | 
| 11 | salunicl.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆) | |
| 12 | 5, 10, 11 | rspcdva 3623 | . 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 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 ωcom 7887 ≼ cdom 8983 SAlgcsalg 46323 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-salg 46324 | 
| This theorem is referenced by: saliunclf 46337 intsal 46345 smfpimbor1lem1 46813 | 
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