Union of Set of Ordinals is Ordinal/Proof 2

Proof
From this, we conclude that $\ds \bigcup A$ is a transitive set.

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


 * $\ds \bigcup A \subseteq A \subseteq \On$

Note that the above applies as well to sets as it does to classes.

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.