Properties of Class of All Ordinals/Zero is Ordinal

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

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


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

Hence $\On$ is a fortiori a superinductive class with respect to 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.