Well-Ordering of Class of All Ordinals under Subset Relation

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

$\On$ is well-ordered by the subset relation such that the following $3$ conditions hold:

Proof
We have that Class of All Ordinals is $g$-Tower.

By Zero is Smallest Ordinal, $0$ is the smallest element of $\On$.

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

The result then follows directly from $g$-Tower is Well-Ordered under Subset Relation.