Axiom:Axiom of Foundation

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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