# 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 | ||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle x \in y\) | \(\iff\) | \(\displaystyle x^+ \in y^+\) |

$\blacksquare$