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 $A$ $\prec$ is a strictly well-founded relation on $A$.

That is, expressed symbolically:


 * ${\prec} \mathrel {\operatorname {We} } A \iff \paren {\prec \operatorname {Or} A \land {\prec} \mathrel {\operatorname {Fr} } A}$

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.