Theorem unisg

Description: The sigma-algebra generated by a collection A is a sigma-algebra on U. A . (Contributed by Thierry Arnoux, 27-Dec-2016)

	Ref	Expression
Assertion	unisg	⊢ ( 𝐴 ∈ 𝑉 → ∪ ( sigaGen ‘ 𝐴 ) = ∪ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		sigagensiga	⊢ ( 𝐴 ∈ 𝑉 → ( sigaGen ‘ 𝐴 ) ∈ ( sigAlgebra ‘ ∪ 𝐴 ) )
2		issgon	⊢ ( ( sigaGen ‘ 𝐴 ) ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ↔ ( ( sigaGen ‘ 𝐴 ) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ ( sigaGen ‘ 𝐴 ) ) )
3	1 2	sylib	⊢ ( 𝐴 ∈ 𝑉 → ( ( sigaGen ‘ 𝐴 ) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ ( sigaGen ‘ 𝐴 ) ) )
4	3	simprd	⊢ ( 𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ ( sigaGen ‘ 𝐴 ) )
5	4	eqcomd	⊢ ( 𝐴 ∈ 𝑉 → ∪ ( sigaGen ‘ 𝐴 ) = ∪ 𝐴 )