Set of Finite Character with Choice Function is of Type M

Theorem

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

Let there exist a choice function for $S$.

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

Let $S$ be according to the hypothesis.

From Class of Finite Character is Closed under Chain Unions:

$S$ is closed under chain unions.

From Closed Set under Chain Unions with Choice Function is of Type $M$:

$S$ is of 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: $\S 5$ Maximal principles