Definition:Strictly Well-Founded Relation

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

Then $\mathcal R$ is a foundational relation on $A$ every non-empty subset of $A$ has an $\mathcal R$-minimal element.

That is, $\mathcal R$ is a foundational relation on $A$ :


 * $\forall s: \left({s \subseteq A \land s \ne \varnothing}\right) \implies \exists y \in s: \forall z \in s: \neg (z \mathrel{\mathcal R} y)$

where $\varnothing$ is the empty set.

Also see

 * Foundational Relation is Antireflexive
 * Foundational Relation is Asymmetric

Special case

 * Definition:Well-Founded Ordered Set