Condition on Proper Lower Sections for Total Ordering to be Well-Ordering/Mistake

Theorem
Let $A$ be a class under a total ordering $\preccurlyeq$.

Let $\preccurlyeq$ be such that:
 * for all proper lower sections $L$ of $A$ under $\preccurlyeq$, there exists a smallest element $x$ of $A$ which is not in $L$.

Then $\preccurlyeq$ is a well-ordering on $A$.

Proof
We already have that $\preccurlyeq$ is a total ordering on $A$.

In order to demonstrate that $\preccurlyeq$ is a well-ordering on $A$, it is necessary to show that every non-empty subset of $A$ has a smallest element.