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:

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

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.

Sources

• Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html