Theorem isrnsigau

Description: The property of being a sigma-algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016)

	Ref	Expression
Assertion	isrnsigau	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ( 𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ ( ∪ 𝑆 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		sgon	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) )
2		elex	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ V )
3		issiga	⊢ ( 𝑆 ∈ V → ( 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) ↔ ( 𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ ( ∪ 𝑆 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) ) )
4	2 3	syl	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ( 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) ↔ ( 𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ ( ∪ 𝑆 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) ) )
5	1 4	mpbid	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ( 𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ ( ∪ 𝑆 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) )