Proof
(related to Proposition: Sets and Their Complements)
 By hypothesis, $A,B$ are sets.
Ad "$(1)$"
 Assume the intersection is empty $A\cap B=\emptyset.$.
 Then, all $x\in A$ are not elements of $B$: $x\not in B$.
 It follows, all $x\in A$ are elements of the complement $x\in B^C.$.
Ad "$(2)$"

"$\Rightarrow$"
 Assume $A$ is a subset of the complement $B^C$: $A\subseteq B^C.$.
 Then, for all $x\in A$ we have $x\not\in B.$.
 Therefore, if $y\in B,$ then surely $y\not in A.$.
 It follows $B\subseteq A^C.$

"$\Leftarrow$"
 Exchange $A$ by $B$ and vice versa in "$\Leftarrow$" to get the other direction of the proof.
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References
Bibliography
 Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016