Axiom:Axiom of Foundation

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


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

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 founded on any set.