Characterisation of Limit Element 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 $A$ such that $x$ is not the smallest element of $A$ under $\preccurlyeq$.

Then:
 * $x$ is a limit element of $A$ under $\preccurlyeq$


 * $x^\prec$ has no greatest element $\preccurlyeq$
 * $x^\prec$ has no greatest element $\preccurlyeq$

where $x^\prec$ denotes the strict lower closure of $x$ in $A$ under $\preccurlyeq$.

Proof
Suppose $x^\prec$ has a greatest element $y$.

Then $x$ is the immediate successor of $y$.

Hence $x$ is not a limit element

Therefore, if $x$ is a limit element, then $x^\prec$ cannot have a greatest element.

Suppose $x$ is an immediate successor.

Then the immediate predecessor of $x$ is the greatest element of $x^\prec$.

Therefore if $x^\prec$ has no greatest element then $x$ cannot be an immediate successor.

Hence if $x$ is not the smallest element of $A$, it must be a limit element.