Transitive Set of Ordinals is Ordinal

Theorem
Let $x$ be a transitive set of ordinals.

Then $x$ is itself an ordinal.

Proof
Let $x$ be a transitive set of ordinals according to the statement of the theorem.

We have from Class of All Ordinals is Well-Ordered by Subset Relation that $\On$ is well-ordered by $\subseteq$.

By Exists Ordinal Greater than Set of Ordinals there exists $\alpha$ such that $\alpha \notin x$.

Hence let $\alpha$ be the smallest ordinal not in $x$.

So all ordinals smaller than $\alpha$ are in $x$.

Hence from Ordinal equals its Initial Segment:


 * $\alpha \subseteq x$

From Transitive Class of Ordinals is Subset of Ordinal not in it:


 * $x \subseteq \alpha$

Hence by set equality:


 * $x = \alpha$

Thus $x$ is an ordinal.