Union of Set of Sets when a Set Intersects All

Theorem
Let $F$ be a set of sets.

Let $S$ be a set.

Suppose that:
 * $\forall A \in F: A \cap S \ne \varnothing$

Then:
 * $\displaystyle F = \bigcup_{x \mathop \in S} \left\{{A \in F: x \in A}\right\}$