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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 | . . 3 ⊢ (𝜑 → ∅ ∈ 𝑆) | |
2 | issald.x | . . . . . . 7 ⊢ 𝑋 = ∪ 𝑆 | |
3 | 2 | eqcomi 2619 | . . . . . 6 ⊢ ∪ 𝑆 = 𝑋 |
4 | 3 | difeq1i 3686 | . . . . 5 ⊢ (∪ 𝑆 ∖ 𝑦) = (𝑋 ∖ 𝑦) |
5 | issald.d | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆) | |
6 | 4, 5 | syl5eqel 2692 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (∪ 𝑆 ∖ 𝑦) ∈ 𝑆) |
7 | 6 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆) |
8 | issald.u | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆) | |
9 | 8 | 3expa 1257 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆) |
10 | 9 | ex 449 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
11 | 10 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
12 | 1, 7, 11 | 3jca 1235 | . 2 ⊢ (𝜑 → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) |
13 | issald.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
14 | issal 39210 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) |
16 | 12, 15 | mpbird 246 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 class class class wbr 4583 ωcom 6957 ≼ cdom 7839 SAlgcsalg 39204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-pw 4110 df-uni 4373 df-salg 39205 |
This theorem is referenced by: salexct 39228 issalnnd 39239 |
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