Properties of Class of All Ordinals/Zero is Ordinal

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

The natural number $0$ is an element of $\On$.

Proof
We have the result that Class of All Ordinals is Minimally Superinductive over Successor Mapping.

Hence $\On$ is a superinductive class  the successor mapping.

Hence, by definition of superinductive class:
 * $\O \in \On$

We identify the natural number $0$ via the von Neumann construction of the natural numbers as:
 * $0 := \O$

and the result follows.