Finite Cartesian Product of Non-Empty Sets is Non-Empty

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Theorem

Let $S_1, S_2, \ldots, S_n$ be non-non-empty sets.


Then their cartesian product $S_1 \times S_2 \times \cdots \times S_n$ is non-empty.


Proof

We use mathematical induction.

The base case $n = 2$ is proved in Kuratowski Formalization of Ordered Pair, and the induction step follows directly from the definition of an ordered tuple.

$\blacksquare$