I am not an an expert in set theory, so this question could be trivial. I am sorry in that case.

Let $I$ be a set and $\{ X_i \}_{i \in I}$ be a collection of sets such that $X_i \neq \emptyset$ for all $i \in I$. The axiom of choice tells precisely that the set $$ \prod_{i \in I} X_i \neq \emptyset $$ is not empty. The use of the word "choice" here is clear. To produce an element $(x_i)\in \prod_{i \in I} X_i$ we need to choose an $x_i \in X_i$ for all $i \in I$.

Now I am wondering: what if there no choice at all? Namely, if $X_i = \{x_i\}$ has one element for each $i \in I$? Indeed, in this case $\prod_{i \in I} X_i$ has only one element, that is $(x_i)$, and there is no choice at all. So the question is the following:

Question: it the axiom of choice needed to prove that the product of a family of sets, each with exactly one element, is not empty?

Thanks!