Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > sigaex | Structured version Visualization version GIF version |
Description: Lemma for issiga 29501 and isrnsiga 29503. 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 29500. (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 2905 | . . 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 | selpw 4115 | . . . . 5 ⊢ (𝑠 ∈ 𝒫 𝒫 𝑜 ↔ 𝑠 ⊆ 𝒫 𝑜) | |
3 | 2 | anbi1i 727 | . . . 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 2726 | . . 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 2632 | . 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 3176 | . . . 4 ⊢ 𝑜 ∈ V | |
7 | pwexg 4776 | . . . 4 ⊢ (𝑜 ∈ V → 𝒫 𝑜 ∈ V) | |
8 | pwexg 4776 | . . . 4 ⊢ (𝒫 𝑜 ∈ V → 𝒫 𝒫 𝑜 ∈ V) | |
9 | 6, 7, 8 | mp2b 10 | . . 3 ⊢ 𝒫 𝒫 𝑜 ∈ V |
10 | 9 | rabex 4740 | . 2 ⊢ {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))} ∈ V |
11 | 5, 10 | eqeltrri 2685 | 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 383 ∧ w3a 1031 ∈ wcel 1977 {cab 2596 ∀wral 2896 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 class class class wbr 4583 ωcom 6957 ≼ cdom 7839 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-rab 2905 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 |
This theorem is referenced by: issiga 29501 isrnsiga 29503 |
Copyright terms: Public domain | W3C validator |