Strict Lower Closure of Limit Element is Infinite

Theorem
Let $A$ be a class.

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

Let $x \in A$ be a limit element of $A$ under $\preccurlyeq$.

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

Then $x^\prec$ is an infinite set.

Proof
Let $x \in A$ be a limit element of $A$ under $\preccurlyeq$.

From Characterisation of Limit Element under Well-Ordering it follows that $x^\prec$ has no greatest element $\preccurlyeq$.

The result follows.