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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigaex | Structured version Visualization version GIF version |
Description: Lemma for issiga 34092 and isrnsiga 34093. The class of sigma-algebras with base set 𝑜 is a set. Note: a more generic version with (𝑂 ∈ V → ...) could be useful for sigaval 34091. (Contributed by Thierry Arnoux, 24-Oct-2016.) |
Ref | Expression |
---|---|
sigaex | ⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3433 | . . 3 ⊢ {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))} = {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} | |
2 | velpw 4609 | . . . . 5 ⊢ (𝑠 ∈ 𝒫 𝒫 𝑜 ↔ 𝑠 ⊆ 𝒫 𝑜) | |
3 | 2 | anbi1i 624 | . . . 4 ⊢ ((𝑠 ∈ 𝒫 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))) ↔ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))) |
4 | 3 | abbii 2806 | . . 3 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} |
5 | 1, 4 | eqtri 2762 | . 2 ⊢ {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} |
6 | vex 3481 | . . . 4 ⊢ 𝑜 ∈ V | |
7 | pwexg 5383 | . . . 4 ⊢ (𝑜 ∈ V → 𝒫 𝑜 ∈ V) | |
8 | pwexg 5383 | . . . 4 ⊢ (𝒫 𝑜 ∈ V → 𝒫 𝒫 𝑜 ∈ V) | |
9 | 6, 7, 8 | mp2b 10 | . . 3 ⊢ 𝒫 𝒫 𝑜 ∈ V |
10 | 9 | rabex 5344 | . 2 ⊢ {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))} ∈ V |
11 | 5, 10 | eqeltrri 2835 | 1 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2105 {cab 2711 ∀wral 3058 {crab 3432 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 𝒫 cpw 4604 ∪ cuni 4911 class class class wbr 5147 ωcom 7886 ≼ cdom 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-pow 5370 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-in 3969 df-ss 3979 df-pw 4606 |
This theorem is referenced by: issiga 34092 isrnsiga 34093 |
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