Subset of Cartesian Product not necessarily Cartesian Product of Subsets

Theorem
Let $A$ and $B$ be sets.

Let $A$ and $B$ both have at least two distinct elements.

Then there exists $W \subseteq A \times B$ such that $W$ is not the cartesian product of a subset of $A$ and a subset of $B$.

Proof
Let $a \in A, b \in A, c \in B, d \in B$ be arbitrary elements of $A$ and $B$.

Let:
 * $W = \left\{{\left({a, c}\right), \left({a, d}\right), \left({b, d}\right)}\right\}$

Then $W \subseteq A \times B$.

Suppose $W = X \times Y$ such that $X \subseteq A, Y \subseteq B$.

Then $a, b \in X$ and $c, d \in Y$.

But $X \times Y$ also contains $\left({b, c}\right)$ which is not in $W$.

Hence the result.