Theorem saluncld

Description: The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	saluncld.1	$$⊢\phi \to S\in \mathrm{SAlg}$$
	saluncld.2	$$⊢\phi \to E\in S$$
	saluncld.3	$$⊢\phi \to F\in S$$
Assertion	saluncld	$$⊢\phi \to E\cup F\in S$$

Proof

Step	Hyp	Ref	Expression
1		saluncld.1	$$⊢\phi \to S\in \mathrm{SAlg}$$
2		saluncld.2	$$⊢\phi \to E\in S$$
3		saluncld.3	$$⊢\phi \to F\in S$$
4		saluncl	$$⊢\left(S\in \mathrm{SAlg}\wedge E\in S\wedge F\in S\right)\to E\cup F\in S$$
5	1 2 3 4	syl3anc	$$⊢\phi \to E\cup F\in S$$