# Equality of Successors

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## 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 |

$\blacksquare$