ThmDex – An index of mathematical definitions, results, and conjectures.
P2904
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 $\emptyset \times X = \emptyset$. $\square$