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


$\blacksquare$


Also see