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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issal | Structured version Visualization version GIF version |
Description: Express the predicate "𝑆 is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
issal | ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2815 | . . 3 ⊢ (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝑆 → 𝑥 = 𝑆) | |
3 | unieq 4924 | . . . . . 6 ⊢ (𝑥 = 𝑆 → ∪ 𝑥 = ∪ 𝑆) | |
4 | 3 | difeq1d 4120 | . . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑥 ∖ 𝑦) = (∪ 𝑆 ∖ 𝑦)) |
5 | 4, 2 | eleq12d 2820 | . . . 4 ⊢ (𝑥 = 𝑆 → ((∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)) |
6 | 2, 5 | raleqbidv 3330 | . . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)) |
7 | pweq 4621 | . . . 4 ⊢ (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆) | |
8 | eleq2 2815 | . . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑦 ∈ 𝑥 ↔ ∪ 𝑦 ∈ 𝑆)) | |
9 | 8 | imbi2d 339 | . . . 4 ⊢ (𝑥 = 𝑆 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) |
10 | 7, 9 | raleqbidv 3330 | . . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) |
11 | 1, 6, 10 | 3anbi123d 1433 | . 2 ⊢ (𝑥 = 𝑆 → ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) |
12 | df-salg 45930 | . 2 ⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))} | |
13 | 11, 12 | elab2g 3668 | 1 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∖ cdif 3944 ∅c0 4325 𝒫 cpw 4607 ∪ cuni 4913 class class class wbr 5153 ωcom 7876 ≼ cdom 8972 SAlgcsalg 45929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rab 3420 df-v 3464 df-dif 3950 df-ss 3964 df-pw 4609 df-uni 4914 df-salg 45930 |
This theorem is referenced by: pwsal 45936 salunicl 45937 saluncl 45938 prsal 45939 saldifcl 45940 0sal 45941 intsal 45951 issald 45954 caragensal 46146 |
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