Zermelo's Well-Ordering Theorem/Converse/Proof 2

Proof
Let $\FF$ be an arbitrary collection of sets.

By assumption all sets can be well-ordered.

Hence the set $\bigcup \FF$ of all elements of sets contained in $\FF$ is well-ordered by some ordering $\preceq$.

By definition then, every subset of $\ds \bigcup \FF$ has a smallest element under $\preceq$.

Also, note that each set in $\FF$ is a subset of $\bigcup \FF$.

Thus, we may define the choice function $C$:
 * $C: \FF \to \bigcup \FF, \map C X = \min X$

where $\min X$ is the smallest element of $X$ under $\preceq$.