Successor is Less than Successor

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Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$.

Then, $x \in y \iff x^+ \in y^+$.


\(\ds x \in y\) \(\implies\) \(\ds x^+ \in y^+\) Subset is Compatible with Ordinal Successor
\(\ds x \in y\) \(\impliedby\) \(\ds x^+ \in y^+\) Sufficient Condition
\(\ds \leadsto \ \ \) \(\ds x \in y\) \(\iff\) \(\ds x^+ \in y^+\)