Is the following proof correct? If so, what proof strategies does it use? If not, can it be fixed? Is the theorem correct?
Proof: Suppose $A \times B \subseteq C \times D$. Let $a$ be an arbitrary element of $A$ and let $b$ be an arbitrary element of $B$. Then $(a, b) \in A \times B$. Since $A \times B \subseteq C \times D$, $(a, b) \in C \times D$. Therefore $a \in C$ and $b \in D$. Since $a$ and $b$ were arbitrary elements of $A$ and $B$, respectively, this shows that $A \subseteq C$ and $B \subseteq D$.
The proof is right, so long as neither $A$ nor $B$ are empty (e.g. if $B$ is empty, we can choose $A$ freely, so the theorem fails; it could be that $A\not\subseteq C$) - you implicitly assume this when you take an element from each.