Theorem difunielsiga

Description: A sigma-algebra is closed under complement relative to its base set. This is immediate from the definition, see issiga , but the library states it nowhere in this form. (Contributed by Vincent Gonzalez, 17-Aug-2026)

	Ref	Expression
Assertion	difunielsiga	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ) → ( ∪ 𝑆 ∖ 𝐴 ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		isrnsigau	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ( 𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ ( ∪ 𝑆 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) ) )
2	1	simprd	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ( ∪ 𝑆 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝒫 𝑆 ( 𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆 ) ) )
3	2	simp2d	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 )
4		difeq2	⊢ ( 𝑥 = 𝐴 → ( ∪ 𝑆 ∖ 𝑥 ) = ( ∪ 𝑆 ∖ 𝐴 ) )
5	4	eleq1d	⊢ ( 𝑥 = 𝐴 → ( ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ↔ ( ∪ 𝑆 ∖ 𝐴 ) ∈ 𝑆 ) )
6	5	rspccva	⊢ ( ( ∀ 𝑥 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑥 ) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ) → ( ∪ 𝑆 ∖ 𝐴 ) ∈ 𝑆 )
7	3 6	sylan	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ) → ( ∪ 𝑆 ∖ 𝐴 ) ∈ 𝑆 )