# Ordinal equals its Initial Segment

## Theorem

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

Let $<$ denote the (strict) usual ordering of $\On$.

Let $\alpha$ be an ordinal.

Then $\alpha$ is equal to its own initial segment:

$\alpha = \set {\beta \in \On: \beta < \alpha}$

## Proof 4

Let $\alpha$ be an ordinal by Definition 4.

$\alpha$ is an ordinal if and only if:

$\alpha$ is an element of every superinductive class.
$\forall \alpha, \beta \in \On: \beta < \alpha \iff \alpha \in \beta$

Hence the statement of the result is equivalent to:

$\alpha = \set {\beta \in \On: \beta \in \alpha}$

which is trivially true by definition of a set.

$\blacksquare$