# Equality of Successors implies Equality of Ordinals

## Theorem

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

Then:

$\forall \alpha, \beta \in \On: \alpha^+ = \beta^+ \implies \alpha = \beta$

## Proof

$\alpha^+$ is the immediate successor of $\alpha$
$\beta^+$ is the immediate successor of $\beta$

and no two distinct elements of $\On$ can have the same immediate successor.

The result follows.

$\blacksquare$