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

$\blacksquare$