Properties of Class of All Ordinals/Superinduction Principle

Theorem
Let $\On$ denote the class of all ordinals.

Let $A$ be a class which satisfies the following $3$ conditions:

That is, let $A$ be a superinductive class under the successor mapping.

Then $A$ contains all ordinals:
 * $\On \subseteq A$

Proof
We note that the zero ordinal, denoted $0$, is identified as the empty set:
 * $0 \:= \O$

Hence by defintion $A$ is indeed a superinductive class under the successor mapping.

By the definition of ordinal:


 * $\alpha$ is an ordinal $\alpha$ is an element of every superinductive class.

Hence $\On$ is a subclass of every superinductive class.

The result follows.