The axioms

• Extensionality, two sets with the same elements are equal.

${\displaystyle \forall x,y,z:(z\in x\iff z\in y)\Rightarrow x=y}$

• Separation, subsets exist

${\displaystyle \forall y_{1},p\,\exists y_{2}:\forall x:x\in y_{2}\iff (p\land x\in y_{1})}$

where ${\displaystyle p}$ is any proposition

• The empty set exists

${\displaystyle \exists x:\forall y:y\notin x}$

• Union, the union of all members of a set is a set.

${\displaystyle \forall x\,\exists y:\forall z:z\in y\iff (\exists u:z\in u\land u\in x)}$

• Power sets exist

${\displaystyle \forall x\,\exists y:\forall z:z\in y\iff (\forall t:t\in z\Rightarrow t\in x)}$

we denote this set ${\displaystyle y}$ by ${\displaystyle P(x)}$

• Infinity, an infinite set exists

${\displaystyle \exists x:(\varnothing \in x)\land (\forall y:y\in x\Rightarrow P(y)\in x)}$

• Foundation, no set is a member of itself

${\displaystyle \forall x:x\neq \varnothing \Rightarrow (\exists y\in x:y\cap x=\varnothing )}$