Set of Subsets of Finite Character of Countable Set is of Type M

Theorem
Let $D$ be a countable set.

Let $S$ be a set of subsets of $D$ such that $S$ is of finite character.

Then $S$ is of type $M$, that is:
 * every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

Proof
We have that $\ds \bigcup S \subseteq D$.

Hence by Subset of Countable Set is Countable, $\ds \bigcup S$ is itself countable.

Hence from Countable Set has Choice Function, $S$ has a choice function.

The result follows from Set of Finite Character with Choice Function is of Type $M$.