Empty Intersection iff Subset of Complement/Corollary/Proof 2

Proof
We have:
 * result $\paren {A \cap C} \cup \paren {B \cap \map \complement C} = \O \iff B \subseteq C \subseteq \map \complement A$:

where the universe $\Bbb U$ is posited.

Let $S$ take the position of $\Bbb U$.

Let $C = X$.

Then we have:
 * $\paren {A \cap X} \cup \paren {B \cap \relcomp S X} = \O \iff B \subseteq X \subseteq \relcomp S A$

Thus we have shown that:
 * $B \subseteq \relcomp S A$

and it follows from Empty Intersection iff Subset of Complement that:
 * $A \cap B = \O$