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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 2834 | . . . . 5 ⊢ ∪ 𝑆 = 𝑋 |
4 | 3 | difeq1i 3951 | . . . 4 ⊢ (∪ 𝑆 ∖ 𝑦) = (𝑋 ∖ 𝑦) |
5 | issald.d | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆) | |
6 | 4, 5 | syl5eqel 2910 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (∪ 𝑆 ∖ 𝑦) ∈ 𝑆) |
7 | 6 | ralrimiva 3175 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆) |
8 | issald.u | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆) | |
9 | 8 | 3expia 1156 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
10 | 9 | ralrimiva 3175 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
11 | issald.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
12 | issal 41325 | . . 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 1448 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ∖ cdif 3795 ∅c0 4144 𝒫 cpw 4378 ∪ cuni 4658 class class class wbr 4873 ωcom 7326 ≼ cdom 8220 SAlgcsalg 41319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-in 3805 df-ss 3812 df-pw 4380 df-uni 4659 df-salg 41320 |
This theorem is referenced by: salexct 41343 issalnnd 41354 |
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