Class of Finite Character is Closed under Chain Unions

Theorem
Let $A$ be a class which has finite character.

Then $A$ is closed under chain unions.

Proof
Let $C$ be a chain of elements of $A$.

$C$ is a set.

To show that $\ds \bigcup C \in A$ it is sufficient to show that every finite subset of $\ds \bigcup C$ is also an element of $A$.

Let $\set {y_1, y_2, \ldots, y_n}$ be an arbitrary subset of $\ds \bigcup C$.

For each $i \le n$, we have that $y_i$ is an element of some $c_i$ of $C$.

Let $c$ be the greatest element by the subset relation of the sets $\set {c_1, c_2, \ldots, c_n}$.

Then each of the sets $y_1, y_2, \ldots, y_n$ is an element of $c$.

Hence $\set {y_1, y_2, \ldots, y_n} \subseteq c$.

Thus every finite subset of $\ds \bigcup C$ is the subset of some element of $C$,

Hence every finite subset of $\ds \bigcup C$ is the subset of some element of $A$.

Hence by Class of Finite Character is Swelled, every finite subset of $\ds \bigcup C$ is an element of $A$.

So, as $A$ has finite character:
 * $\ds \bigcup C \in A$