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 5071 | . . . 4 ⊢ (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω)) | |
3 | unieq 4851 | . . . . 5 ⊢ (𝑦 = 𝑇 → ∪ 𝑦 = ∪ 𝑇) | |
4 | 3 | eleq1d 2899 | . . . 4 ⊢ (𝑦 = 𝑇 → (∪ 𝑦 ∈ 𝑆 ↔ ∪ 𝑇 ∈ 𝑆)) |
5 | 2, 4 | imbi12d 347 | . . 3 ⊢ (𝑦 = 𝑇 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆) ↔ (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆))) |
6 | salunicl.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
7 | issal 42606 | . . . . . 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 234 | . . . 4 ⊢ (𝜑 → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) |
10 | 9 | simp3d 1140 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
11 | salunicl.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆) | |
12 | 5, 10, 11 | rspcdva 3627 | . 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 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∖ cdif 3935 ∅c0 4293 𝒫 cpw 4541 ∪ cuni 4840 class class class wbr 5068 ωcom 7582 ≼ cdom 8509 SAlgcsalg 42600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-salg 42601 |
This theorem is referenced by: saliuncl 42614 intsal 42620 smfpimbor1lem1 43080 |
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