Union of Ordinal is Subset of Itself

Theorem
Let $\alpha$ be an ordinal.

Then:
 * $\bigcup \alpha \subseteq \alpha$

where $\bigcup \alpha$ denotes the union of $\alpha$.

Proof
Let $x \in \bigcup \alpha$.

Then:
 * $\exists \beta \in \alpha: x \in \beta$

By Element of Ordinal is Ordinal, $\beta$ is an ordinal.

Thus:
 * $x \in \beta$ and $\beta \in \alpha$

Hence by Ordinal Membership is Transitive:
 * $x \in \alpha$