# Class of All Ordinals is Well-Ordered by Subset Relation/Proof 2

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## Theorem

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

Then the restriction of the subset relation, $\subseteq$, to $\On$ is a well-ordering.

That is:

- $\subseteq$ is an ordering on $\On$.
- If $A$ is a non-empty subclass of $\On$, then $A$ has a smallest element under the subset relation.

## Proof

From Class of All Ordinals is $g$-Tower, $\On$ is a $g$-tower.

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

$\blacksquare$

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.3$