Union of Chain in Set of Finite Character with Countable Union is Maximal Element/Proof

Theorem

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

Let its union $\ds \bigcup S$ be countable.

Then $\ds \bigcup S$ is a maximal element of $S$ under the subset relation.

Proof

Let $S$ be as by hypothesis.

Aiming for a contradiction, suppose $\ds \bigcup S$ is not a maximal element of $S$ under the subset relation.

Then:

$\exists T \subseteq S: \ds \bigcup S \subsetneqq T$

Thus:

$\exists x \in S: x \ne \ds \bigcup S$

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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: Exercise $6.1$