Ordinal is Transitive/Proof 4

Proof
Let $\alpha$ be an ordinal by Definition 4.

The proof proceeds by the Principle of Superinduction.

From Empty Class is Transitive we start with the fact that $0$ is transitive.

Let $x$ be transitive.

From Successor Set of Transitive Set is Transitive:
 * $x^+$ is transitive.

We have that Class is Transitive iff Union is Subclass.

Hence the union of a chain of transitive sets is transitive.

Hence the result by the Principle of Superinduction.