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
\(\ds x = y\) | \(\leadsto\) | \(\ds x^+ = y^+\) | Substitutivity of Equality |
Conversely,
\(\ds x^+ = y^+\) | \(\leadsto\) | \(\ds \bigcup x^+ = \bigcup y^+\) | Substitutivity of Equality | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds x = y\) | Union of Successor Ordinal |
$\blacksquare$