Definition:Strictly Well-Founded Relation

Definition
Let $\left({A, \prec}\right)$ be a relational structure where $A$ is either a proper class or a set.

Then $\prec$ is a foundational relation on $A$ iff every nonempty subset of $A$ has a minimal element.

That is, $\prec$ is a foundational relation on $A$ iff:


 * $\forall x: \left({x \subseteq A \land x \ne \varnothing}\right) \implies \exists y \in x: \forall z \in x: \neg z \prec y$

where $\varnothing$ is the empty set.

Also see

 * Foundational Relation is Antireflexive
 * Foundational Relation is Asymmetric