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

Theorem
Let $\left({C, \le}\right)$ be a well-ordered class.

Let $x \in C$.

Suppose that $x$ is maximal in $C$.

Then $x$ has an immediate successor 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 $S$, $z \in S$.

This contradicts the minimality of $y$.

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