Zermelo's Well-Ordering Theorem/Converse/Proof 1

Proof
Let $S$ be an arbitrary set.

By assumption $S$ is well-orderable.

Let $\preccurlyeq$ be a well-ordering on $S$.

Let $T \subseteq S$ be an arbitrary non-empty subset of $S$.

As $S$ is a well-ordered set, $T$ has a unique smallest element by $\preccurlyeq$.

Thus, we may define the choice function $C: \powerset S \setminus \set \O \to S$ as:
 * $\forall T \in \powerset S \setminus \set \O: \map C T$ is the smallest element of $T$ under $\preccurlyeq$