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 $\preccurlyeq$.