Axiom:Axiom of Foundation

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

That is:
 * $\forall S: \paren {\paren {\exists x: x \in S} \implies \exists y \in S: \forall z \in S: \neg \paren {z \in y} }$

The antecedent states that $S$ is not empty.

Also defined as
It can also be stated as:


 * For every non-empty set $S$, there exists an element $x \in S$ such that $x$ and $S$ are disjoint.


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


 * A set contains a (membership) minimal element.


 * The membership relation is a foundational relation on any non-empty set.

Also known as
The axiom of foundation is also known as the axiom of regularity.