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 $S$ be such that:
 * $\forall x \in S: x$ is 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