Transitive Class of Ordinals is Subset of Ordinal not in it

Theorem
Let $A$ be a transitive class of ordinals.

Let $\alpha$ be an ordinal which is not an element of $A$.

Then:
 * $A \subseteq \alpha$

Proof
Let $A$ and $\alpha$ be.

Let $\beta \in A$ be arbitrary.

Because $\beta \in A$ and $\alpha \notin A$ we have that:
 * $\beta \ne \alpha$

Because $\beta \in A$ and $A$ is transitive:
 * all elements of $\beta$ are in $A$

But because $\alpha \notin A$:
 * $\alpha \notin \beta$

Thus we have:
 * $\alpha \notin \beta$

and:
 * $\alpha \ne \beta$

Hence from Ordinal Membership is Trichotomy:
 * $\beta \in \alpha$

As $\beta$ is arbitrary, it follows that:
 * $A \subseteq \alpha$