Successor is Less than Successor

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

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

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

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

Sufficient Condition
Suppose $y^+ \in x^+$.

By the definition of successor, $y^+ \in x \lor y^+ = x$.

Suppose $y^+ = x$.

By Ordinal Less than Successor, $y \in x$.

Suppose $y^+ \in x$.

By Ordinal Less than Successor, $y \in y^+$.

By Ordinal is Transitive, $y \in x$.