Set of Finite Character with Choice Function is Type M

Theorem

Let $S$ be a set of sets of finite character.

Let $S$ have a choice function $C$ for its union $\ds \bigcup S$.

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

By Class of Finite Character is Swelled, a set of finite character is swelled.

By Class of Finite Character is Closed under Chain Unions a set of finite character is closed under chain unions.

The result follows from Swelled Set which is Closed under Chain Unions with Choice Function is Type $M$.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 6$ Another approach to maximal principles: Corollary $6.4$