Axiom:Axiom of Foundation

Axiom
For all non-empty sets, there is an element of the set that shares no element with the set.


 * $\forall S: \left({ \exists x: x \in S \implies \exists y \in S: \forall x \in S: \neg x \in y }\right)$

The antecedent states that $S$ is not empty.

Otherwise known as the Axiom of Regularity.

It can also be stated as:


 * A set contains no infinitely descending (membership) sequence.


 * A set contains a (membership) minimal element.


 * The membership relation is a foundational relation on any set.