Definition:Strict Well-Ordering

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Definition 1

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

Then $\prec$ is a strict well-ordering on $A$ if and only if $\prec$ is a foundational relation on $A$.

That is, expressed symbolically:

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

Definition 2

Let $A$ be a set or class.

Let $\prec$ be a relation on $A$.

Then $\prec$ is a strict well-ordering of $A$ if and only if:

$\prec$ connects $A$
$\prec$ is well-founded. That is, whenever $b$ is a non-empty subset of $A$, $b$ has a $\prec$-minimal element.

Also see