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Mirrors > Home > MPE Home > Th. List > Mathboxes > issald | Structured version Visualization version GIF version |
Description: Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
issald.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
issald.z | ⊢ (𝜑 → ∅ ∈ 𝑆) |
issald.x | ⊢ 𝑋 = ∪ 𝑆 |
issald.d | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆) |
issald.u | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆) |
Ref | Expression |
---|---|
issald | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issald.z | . 2 ⊢ (𝜑 → ∅ ∈ 𝑆) | |
2 | issald.x | . . . . . 6 ⊢ 𝑋 = ∪ 𝑆 | |
3 | 2 | eqcomi 2740 | . . . . 5 ⊢ ∪ 𝑆 = 𝑋 |
4 | 3 | difeq1i 4098 | . . . 4 ⊢ (∪ 𝑆 ∖ 𝑦) = (𝑋 ∖ 𝑦) |
5 | issald.d | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆) | |
6 | 4, 5 | eqeltrid 2836 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (∪ 𝑆 ∖ 𝑦) ∈ 𝑆) |
7 | 6 | ralrimiva 3145 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆) |
8 | issald.u | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆) | |
9 | 8 | 3expia 1121 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
10 | 9 | ralrimiva 3145 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
11 | issald.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
12 | issal 44708 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) |
14 | 1, 7, 10, 13 | mpbir3and 1342 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 ∖ cdif 3925 ∅c0 4302 𝒫 cpw 4580 ∪ cuni 4885 class class class wbr 5125 ωcom 7822 ≼ cdom 8903 SAlgcsalg 44702 |
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 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rab 3419 df-v 3461 df-dif 3931 df-in 3935 df-ss 3945 df-pw 4582 df-uni 4886 df-salg 44703 |
This theorem is referenced by: salexct 44728 issalnnd 44739 |
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