Condition for Total Ordering to be Well-Ordering

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

Then $\preccurlyeq$ is a well-ordering on $A$ the following $3$ conditions are fulfilled:


 * $(1): \quad A$ has a smallest element $x_0$


 * $(2): \quad$ Every element of $A$ except the greatest (if there is one) has an immediate successor


 * $(3): \quad$ For every proper lower section $L$ of $A$, if $L$ has no greatest element, then there is a smallest element of $A$ which is not in $L$.