Principle of Finite Choice

Theorem
Let $\mathbb S$ be a set of sets such that:
 * $\forall S \in \mathbb S: S \ne \varnothing$

that is, none of the sets in $\mathbb S$ may be empty.

Let $\mathbb S$ be finite.

Then there exists a choice function $f: \mathbb S \to \bigcup \mathbb S$ defined as:
 * $\forall S \in \mathbb S: \exists x \in S: f \left({S}\right) = x$

Thus, if $\mathbb S$ is finite, we can construct a choice function on $\mathbb S$ by picking one element from each member of $\mathbb S$.