# Successor is Less than Successor

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## Theorem

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

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

## Proof

 $\displaystyle x \in y$ $\implies$ $\displaystyle x^+ \in y^+$ Subset is Compatible with Ordinal Successor $\displaystyle x \in y$ $\impliedby$ $\displaystyle x^+ \in y^+$ Sufficient Condition $\displaystyle \implies \ \$ $\displaystyle x \in y$ $\iff$ $\displaystyle x^+ \in y^+$

$\blacksquare$