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

Theorem

Let $C$ be a well-ordered class under an ordering $\le$.

Let $x \in C$.

Suppose that $x$ is not the greatest element in $C$.

Then $x$ has an immediate successor element in $C$.

Proof

Let $x$ be an element of $C$ which is not the greatest element of $C$.

Let $S$ be the class of successor elements of $x$ in $C$.

We have that $S$ is a subclass of $C$.

Also, $S$ is non-empty because $x$ is not the greatest element.

Thus $S$ is a non-empty subclass of $C$.

We have by hypothesis that $\le$ is a well-ordering on $C$

Hence $S$ has a smallest element, $y$.

By the definition of $S$:

$x < y$

Aiming for a contradiction, suppose that for some $z \in C$, $x < z < y$.

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

This contradicts the fact that $y$ is the smallest element of $S$.

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

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Proposition $1.4$