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

Proof
Let $S$ be as.

$\ds \bigcup S$ is not a maximal element of $S$ under the subset relation.

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

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