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

 * $\displaystyle

\begin{align*} x \in \bigcup A &\implies \exists y \in A: x \in y \\ &\implies \exists y \subseteq \bigcup A: x \subseteq y \\ &\implies x \subseteq \bigcup A \end{align*} $

From this, we conclude that $\bigcup A$ is a transitive class. It also 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 because it is a transitive set and is well-ordered by $\Epsilon$.