ThmDex – An index of mathematical definitions, results, and conjectures.

Finite cartesian product with empty set is empty

Formulation 0

Let $\emptyset$ be the D13: Empty set.
Let $X_1, \ldots, X_N$ each be a D11: Set such that

(i)	\begin{equation} \exists \, n \in 1, \ldots, N : X_n = \emptyset \end{equation}

Then \begin{equation} \prod_{n = 1}^N X_n = \emptyset \end{equation}

Proofs

Proof 0

Let $\emptyset$ be the D13: Empty set.
Let $X_1, \ldots, X_N$ each be a D11: Set such that

(i)	\begin{equation} \exists \, n \in 1, \ldots, N : X_n = \emptyset \end{equation}

This result is a particular case of R4263: Countable cartesian product with empty set is empty. $\square$