By definition of a
D191: Binary cartesian set product, we have
\begin{equation}
X \times \emptyset
= \{ (x, y) : x \in X, y \in \emptyset \}
\end{equation}
The predicate expression $x \in X, y \in \emptyset$ is false for any $(x, y)$, whence $X \times \emptyset = \emptyset$. Since $y \in \emptyset, x \in X$ is also false for any $(x, y)$, then also $\emptyset \times X = \emptyset$. $\square$
