Non-Greatest Element of Well-Ordered Class has Immediate Successor

Theorem
Let $\left({C,\le}\right)$ be a Definition:Well-Ordered Class.

Let $x \in C$ and suppose that $x$ is maximal in $C$.

Then $x$ has a Definition:Immediate Successor Element in $C$.

Proof
Let $x$ be a non-maximal element of $C$.

Let $S$ be the class of successors of $x$ in $C$. $S$ is non-empty because $x$ is not maximal.

By Proper Well-Ordering Determines Smallest Elements, $S$ has a minimal element, $y$.

Then $x < y$ by the definition of $S$.

Suppose that for some $z \in C$, $x < z < y$.

Then by the definition of $S$, $z \in S$, contradicting the minimality of $y$.

Thus $y$ is the immediate successor element of $x$.