Definition:Strictly Well-Founded Relation

Definition
Let $\struct {A, \RR}$ be a relational structure where $A$ is either a proper class or a set.

Then $\RR$ is a strictly well-founded relation on $A$ every non-empty subset of $A$ has an $\RR$-minimal element.

That is, $\RR$ is a strictly well-founded relation on $A$ :


 * $\forall s: \paren {s \subseteq A \land s \ne \O} \implies \exists y \in s: \forall z \in s: \neg \paren {z \mathrel \RR y}$

where $\O$ is the empty set.

Also see

 * Strictly Well-Founded Relation is Antireflexive
 * Strictly Well-Founded Relation is Asymmetric

Special case

 * Definition:Well-Founded Ordered Set