Theorem sssigagen

Description: A set is a subset of the sigma-algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017)

	Ref	Expression
Assertion	sssigagen	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ⊆ ( sigaGen ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ssintub	⊢ 𝐴 ⊆ ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 }
2		sigagenval	⊢ ( 𝐴 ∈ 𝑉 → ( sigaGen ‘ 𝐴 ) = ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } )
3	1 2	sseqtrrid	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ⊆ ( sigaGen ‘ 𝐴 ) )