Theorem sigaex

Description: Lemma for issiga and isrnsiga . The class of sigma-algebras with base set o is a set. Note: a more generic version with ( O e. _V -> ... ) could be useful for sigaval . (Contributed by Thierry Arnoux, 24-Oct-2016)

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

Proof

Step	Hyp	Ref	Expression
1		df-rab	⊢ { 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	⊢ ( 𝑠 ∈ 𝒫 𝒫 𝑜 ↔ 𝑠 ⊆ 𝒫 𝑜 )
3	2	anbi1i	⊢ ( ( 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	⊢ { 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	⊢ { 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	⊢ 𝑜 ∈ V
7		pwexg	⊢ ( 𝑜 ∈ V → 𝒫 𝑜 ∈ V )
8		pwexg	⊢ ( 𝒫 𝑜 ∈ V → 𝒫 𝒫 𝑜 ∈ V )
9	6 7 8	mp2b	⊢ 𝒫 𝒫 𝑜 ∈ V
10	9	rabex	⊢ { 𝑠 ∈ 𝒫 𝒫 𝑜 ∣ ( 𝑜 ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝑠 ( 𝑜 ∖ 𝑥 ) ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝒫 𝑠 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠 ) ) } ∈ V
11	5 10	eqeltrri	⊢ { 𝑠 ∣ ( 𝑠 ⊆ 𝒫 𝑜 ∧ ( 𝑜 ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝑠 ( 𝑜 ∖ 𝑥 ) ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝒫 𝑠 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠 ) ) ) } ∈ V