Categories of Elements under Well-Ordering

Theorem
Let $A$ be a class

Let $\preccurlyeq$ be a well-ordering on $A$.

Let $x \in A$ be an element of $A$.

Then $x$ falls into one of the following $3$ categories:


 * $(1): \quad x$ is the smallest element of $A$ $\preccurlyeq$


 * $(2): \quad x$ is the immediate successor of another element $y \in A$ $\preccurlyeq$


 * $(3): \quad x$ is a limit element of $A$ under $\preccurlyeq$.

Proof
Let $x \in A$.


 * $(1): \quad x$ is the smallest element of $A$ $\preccurlyeq$:

We note that $\preccurlyeq$ is a well-ordering on $A$.

Hence as $A \subseteq A$ we have that $A$ has a smallest element.

Hence there exists one element of $A$ which is that smallest element of $A$.

Let $x$ be that smallest element of $A$.

By definition of smallest element:
 * $\forall y \in A: x \preccurlyeq y$

Hence $x$ cannot be the immediate successor of another element $y \in A$ $\preccurlyeq$.


 * $(2): \quad x$ is the immediate successor of another element $y \in A$ $\preccurlyeq$:

Let $y \in A$ such that $y$ is not the greatest element of $A$ $\preccurlyeq$.

Then:
 * $\forall y \in A: \exists x \in A: y \preccurlyeq x$

such that $x$ is an immediate successor of $y$.

By definition of immediate successor, it follows that $x$ is not the smallest element of $A$ $\preccurlyeq$.


 * $(3): \quad x$ is a limit element of $A$ under $\preccurlyeq$:

Suppose there exists $x$ such that:
 * $x$ is not the smallest element of $A$ $\preccurlyeq$

and:
 * $x$ is not the immediate successor of another element $y \in A$ $\preccurlyeq$.

Then by definition $x$ is a limit element of $A$ under $\preccurlyeq$.

The result follows.