Intersection of Set whose Every Element is Closed under Chain Unions is also Closed under Chain Unions

Theorem
Let $S$ be a set of sets.

Let $x \in S$ be closed under chain unions.

Then the intersection $\ds \bigcap S$ of $S$ is also closed under chain unions.

Also see

 * Intersection of Set whose Every Element is Closed under Mapping is also Closed under Mapping