Ordinal is Transitive

Theorem
Let $A$ be an ordinal. Then $A$ is a transitive class.

That is, every ordinal is a transitive class.

Proof
By definition $A$ is an ordinal iff:
 * $\forall x \in A: \left\{{y \in A : y \subset x}\right\} = x$

Thus:
 * $\forall x \in A: \forall y: \left({\left({y \in A \land y \subset x}\right) \iff y \in x}\right)$

The biconditional can be reduced to an implication, and therefore:
 * $\forall x \in A: \forall y \in x: \left({y \in A \land y \subset x}\right)$

By applying the Rule of Simplification on the $\land$ statement:
 * $\forall x \in A: \forall y \in x: y \in A$

By the definition of a subset:
 * $\forall x \in A: x \subseteq A$

Finally, by the definition of a Transitive Class, $Tr A$.