Theorem saldifcld

Description: The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	saldifcld.1	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
	saldifcld.2	⊢ ( 𝜑 → 𝐸 ∈ 𝑆 )
Assertion	saldifcld	⊢ ( 𝜑 → ( ∪ 𝑆 ∖ 𝐸 ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		saldifcld.1	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
2		saldifcld.2	⊢ ( 𝜑 → 𝐸 ∈ 𝑆 )
3		saldifcl	⊢ ( ( 𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ) → ( ∪ 𝑆 ∖ 𝐸 ) ∈ 𝑆 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ∪ 𝑆 ∖ 𝐸 ) ∈ 𝑆 )