(A cap C) cup (B cap Complement C) = Empty iff B subset C subset Complement A

Theorem
Let $A$, $B$ and $C$ be subsets of a universe $\Bbb U$.

Then:
 * $\paren {A \cap C} \cup \paren {B \cap \map \complement C} = \O \iff B \subseteq C \subseteq \map \complement A$

where $\map \complement C$ denotes the complement of $C$ in $\Bbb U$.