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Theorem sigaex 29953
Description: Lemma for issiga 29955 and isrnsiga 29957. 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 29954. (Contributed by Thierry Arnoux, 24-Oct-2016.)
Assertion
Ref Expression
sigaex {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V
Distinct variable group:   𝑜,𝑠

Proof of Theorem sigaex
StepHypRef Expression
1 df-rab 2916 . . 3 {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))} = {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))}
2 selpw 4137 . . . . 5 (𝑠 ∈ 𝒫 𝒫 𝑜𝑠 ⊆ 𝒫 𝑜)
32anbi1i 730 . . . 4 ((𝑠 ∈ 𝒫 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))) ↔ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))))
43abbii 2736 . . 3 {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))}
51, 4eqtri 2643 . 2 {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))}
6 vex 3189 . . . 4 𝑜 ∈ V
7 pwexg 4810 . . . 4 (𝑜 ∈ V → 𝒫 𝑜 ∈ V)
8 pwexg 4810 . . . 4 (𝒫 𝑜 ∈ V → 𝒫 𝒫 𝑜 ∈ V)
96, 7, 8mp2b 10 . . 3 𝒫 𝒫 𝑜 ∈ V
109rabex 4773 . 2 {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))} ∈ V
115, 10eqeltrri 2695 1 {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1987  {cab 2607  wral 2907  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  ωcom 7012  cdom 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-pow 4803
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132
This theorem is referenced by:  issiga  29955  isrnsiga  29957
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