# Successor is Less than Successor

## 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

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

$\blacksquare$