Properties of Class of All Ordinals/Union of Chain of Ordinals is Ordinal

Theorem
Let $\On$ denote the class of all ordinals.

Let $C$ be a chain of elements of $\On$.

Then its union $\ds \bigcup C$ is also an element of $\On$.

Proof
We have the result that Class of All Ordinals is Minimally Superinductive over Successor Mapping.

Hence $\On$ is a superinductive class  the successor mapping.

Hence, by definition of superinductive class:
 * $\On$ is closed under chain unions.

That is:
 * $\forall C \in \On: \ds \bigcup C \in \On$

where:
 * $C$ is a chain of elements of $\On$
 * $\ds \bigcup C$ is the union of $C$.