Equality of Successors

Theorem

Let $x$ and $y$ be ordinals.

Let $x^+$ denote the successor set of $x$.

Then, $x = y \iff x^+ = y^+$

Proof

 $\displaystyle x = y$ $\implies$ $\displaystyle x^+ = y^+$ Substitutivity of Equality

Conversely,

 $\displaystyle x^+ = y^+$ $\implies$ $\displaystyle \bigcup x^+ = \bigcup y^+$ Substitutivity of Equality $\displaystyle$ $\implies$ $\displaystyle x = y$ Union of Successor Ordinal

$\blacksquare$