# 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:

 $(1)$ $:$ the smallest ordinal is $0$ $(2)$ $:$ for $\alpha \in \On$, the immediate successor of $\alpha$ is its successor set $\alpha^+$ $(3)$ $:$ every limit ordinal is the union of the set of smaller ordinals.

## 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.

$\blacksquare$