Zermelo–Fraenkel set theory

Zermelo–Fraenkel set theory with the Axiom of Choice or ZFC is a popular first-order axiomatic set theory.

Axiom of Extensionality
Two sets with the same elements are equal.

\[\forall x: \forall y: [\forall z: (z \in x \Leftrightarrow z \in y) \Rightarrow (x = y)]\]

Axiom of Pairing
Given two sets, there is a third one containing only those two.

\[\forall x: \forall y: \exists z: \forall w: [w \in z \Rightarrow (w = x \vee w = y)]\]

Axiom of Regularity
Sets may not contain themselves indirectly or directly.

\[\forall x: [\exists a: (a \in x) \Rightarrow \exists y: (y \in x \wedge \not \exists z: (z \in y \wedge z \in x))]\]