# Zero is Smallest Ordinal

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## Theorem

The natural number $0$ is the smallest ordinal.

## Proof

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

By Zero is Ordinal, $0$ is an element of $\On$.

We identify the natural number $0$ via the von Neumann construction of the natural numbers as:

- $0 := \O$

By Empty Class is Subclass of All Classes:

- $\forall \alpha \in \On: \O \subseteq \alpha$

Hence the result by definition of smallest element.

$\blacksquare$