Theorem unielsiga

Description: A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017) (Shortened by Thierry Arnoux, 6-Jun-2017.)

	Ref	Expression
Assertion	unielsiga	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		sgon	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) )
2		baselsiga	⊢ ( 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) → ∪ 𝑆 ∈ 𝑆 )
3	1 2	syl	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆 )