Theorem sigaex 34111

Description: Lemma for issiga 34113 and isrnsiga 34114. 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 34112. (Contributed by Thierry Arnoux, 24-Oct-2016.)

Assertion

Ref	Expression
sigaex	⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V

Distinct variable group:   𝑜,𝑠

Proof of Theorem sigaex

Step	Hyp	Ref	Expression
1		df-rab 3437	. . 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 4605	. . . . 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 2809	. . 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 2765	. 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 3484	. . . 4 ⊢ 𝑜 ∈ V
7		pwexg 5378	. . . 4 ⊢ (𝑜 ∈ V → 𝒫 𝑜 ∈ V)
8		pwexg 5378	. . . 4 ⊢ (𝒫 𝑜 ∈ V → 𝒫 𝒫 𝑜 ∈ V)
9	6, 7, 8	mp2b 10	. . 3 ⊢ 𝒫 𝒫 𝑜 ∈ V
10	9	rabex 5339	. 2 ⊢ {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))} ∈ V
11	5, 10	eqeltrri 2838	1 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  {cab 2714  ∀wral 3061  {crab 3436  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907   class class class wbr 5143  ωcom 7887   ≼ cdom 8983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-pow 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602

This theorem is referenced by:  issiga  34113  isrnsiga  34114