Theorem elsigass

Description: An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017)

	Ref	Expression
Assertion	elsigass	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ) → 𝐴 ⊆ ∪ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		sgon	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) )
2		sigasspw	⊢ ( 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) → 𝑆 ⊆ 𝒫 ∪ 𝑆 )
3	1 2	syl	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ 𝒫 ∪ 𝑆 )
4	3	sselda	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ) → 𝐴 ∈ 𝒫 ∪ 𝑆 )
5	4	elpwid	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ) → 𝐴 ⊆ ∪ 𝑆 )