Empty Intersection iff Subset of Relative Complement

Theorem
Let $S$ be a set.

Let $A, B$ be subset of $S$.

Then $A \cap B = \O \iff A \subseteq \relcomp S B$

Proof

 * $A \cap B = \O$


 * $\forall x \in S: x \notin A \cap B$ by Empty Set as Subset


 * $\forall x \in S: \neg \paren {x \in A \land x \in B}$ by definition of intersection


 * $\forall x \in S: x \notin A \lor x \notin B$ by De Morgan's Laws (Logic)/Disjunction of Negations


 * $\forall x \in S: x \in A \implies x \notin B$ by Rule of Material Implication


 * $\forall x \in S: x \in A \implies x \in \relcomp S B$ by definition of relative complement


 * $A \subseteq \relcomp S B$ by Subset in Subsets.