# Axiom:Axiom of Foundation (Classes)

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## Axiom

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

- $\forall S: \neg \paren {S = z: \forall y: \paren {\neg \paren {y \in z} } } \implies \exists x \in S: \neg \paren {\exists w: w \in S \land w \in x}$

## Also known as

Otherwise known as the **Axiom of Regularity**.

## Also defined as

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.