# Axiom of Regularity

<https://www.wikiwand.com/en/Axiom_of_regularity>

**Axiom of regularity** or **axiom of foundation** states that every non-empty set $$x$$ contains a member $$y$$ such that $$x$$ and $$y$$ are disjoint sets.

$$
\forall x\[
\exists a(a\in x)
\Rightarrow
\exists y (
y\in x
\land
\lnot \exists z(z\in y\land z\in x)
)
]
$$

This implies, for example, that no set is an element of itself and that every set has an ordinal rank.