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
From Class of All Ordinals is Well-Ordered by Subset Relation:
 * $\alpha^+$ 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.