Union of Set of Ordinals is Ordinal

Theorem
Let $A$ be a class of ordinals. That is, $A \subseteq \operatorname{On}$, where $\operatorname{On}$ denotes the ordinal class.

Then $\bigcup A$ is an ordinal.

Proof
From this, we conclude that $\displaystyle \bigcup A$ is a transitive class.

From Class is Transitive iff Union is Subset, it follows that:


 * $\displaystyle \bigcup A \subseteq A \subseteq \operatorname{On}$

By Subset of Well-Ordered Set is Well-Ordered, $A$ is also well-ordered by $\Epsilon$.

Thus by Alternative Definition of Ordinal, $\bigcup A$ is an ordinal.