Empty Intersection iff Subset of Complement/Corollary

Corollary to Intersection is Subset of Union of Intersections with Complements
Let $A, B, S$ be sets such that $A, B \subseteq S$.

Then:
 * $\exists X \in \powerset S: \paren {A \cap X} \cup \paren {B \cap \overline X} = \O \iff A \cap B = \O$

where $\overline X$ denotes the complement of $X$.

Proof
Let there exist such a set $X$.

Then: