Successor is Less than Successor

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

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

Proof
The last part is a contradiction, so $y^+ \notin x^+$.

By Ordinal Membership Trichotomy, $x^+ \in y^+$.