Definition:Strict Well-Ordering/Definition 1

Definition
Let $\prec$ be a strict total ordering on a class $A$.

Then $\prec$ is a strict well-ordering on $A$ iff $\prec$ is a foundational relation on $A$.

That is, expressed symbolically:


 * ${\prec} \mathrel{\operatorname{We}} A \iff \left({\prec \operatorname{Or} A \land {\prec} \mathrel{\operatorname{Fr}} A}\right)$

Also see

 * Definition:Well-Ordering

Note
Note that our definition does not require $A$ to be a set, as it can also be a proper class. This allows ordinals to be expressed in terms of strict well-orderings on $\in$, as the class of all ordinal numbers is a proper class.