Swelled Set which is Closed under Chain Unions with Choice Function is Type M

Theorem
Let $S$ be a set of sets which:
 * is closed under chain unions
 * has a choice function $C$ for its union $\ds \bigcup S$.

Then:
 * $S$ is swelled


 * $S$ is of type $M$.
 * $S$ is of type $M$.

Sufficient Condition
Let $S$ be swelled.

Let $b \in S$ be arbitrary.

From Closure under Chain Unions with Choice Function implies Elements with no Immediate Extension:


 * $b$ is the subset of an element of $S$ which has no immediate extension in $S$.

Let $x \in S$ have no immediate extension in $S$.

Then from Element of Swelled Set with no Immediate Extension is Maximal, $x$ is a maximal element under the subset relation on $S$.

That is:
 * $b$ is the subset of a maximal element of $S$ under the subset relation.

As $b$ is arbitrary:
 * every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

Hence, by definition, $S$ is of type $M$.

Necessary Condition
Let $S$ be of type $M$.