Axiom:Axiom of Foundation (Classes)

For any non-empty class, there is an element of the class that shares no element with the class.


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

Otherwise known as the Axiom of Regularity.

It can also be stated as:


 * A class contains no infinitely descending (membership) sequence. (only equivalent with AoC)


 * A class contains a (membership) minimal element.