I have already tried to prove it in my own way and would like to share my results in hope that a mathematician somewhere can tell if I am correct. I believe that the proof consists of 2 parts: First, prove left to right that $A \subseteq B \implies B' \subseteq A'$. Second, prove right to left that $B' \subseteq A' \implies A \subseteq B$.
In the first part, we assume $A \subseteq B$. We have $x \in A' \implies x \not \in A$ and since $A \subseteq B$, then $x \not \in B$. On the other hand, $x \in B' \implies x \not \in B$ and $x \in A'$ therefore $A' \cap B' = B'$ which means $B' \subseteq A'$.
The second part of the proof is more or less the same. Is my approach correct? Thanks.