Axiom of Regularity
Axioms
$$ \forall X \left( \exists x_{0} ( x_{0} \in X ) \implies \exists y ( y \in X \land \lnot \exists x ( x \in y \land x \in X )) \right) $$ Every set $X \ne \emptyset$ has an element that is mutually exclusive with itself.
$$ \forall X \left( \exists x_{0} ( x_{0} \in X ) \implies \exists y ( y \in X \land \lnot \exists x ( x \in y \land x \in X )) \right) $$ Every set $X \ne \emptyset$ has an element that is mutually exclusive with itself.