So I worked out the following:
$A \times B = \{(a,b) | a \in A \land b \in B\} $, since $A \times B = A \times C$ then the following is true:
$A \times C = \{(a,b) | a \in A \land b \in C\} $, therefore $\forall b(b \in B \rightarrow b \in C) \equiv B \subseteq C$
and that without loss of generality $C \subseteq B \therefore B = C$.
But I'm not clear on how this proves that $B \subseteq C$ vs. $B \subset C?$
Best Answer
Assume $A$ is not empty and $A\times B = A \times C$.
Let $a$ be an element of $A$.
If $x \in B$, then $(a,x) \in A\times B$.
So $(a,x) \in A\times C$, hence $x \in C$.
Thus $B \subseteq C$.
Similarly $C \subseteq B$.