Set theory is an extensive topic introduced elsewhere. It can be written as a first order theory.
There is one axiom schema, Abstraction (or Comprehension), which can generate infinitely many axioms
Axiom Schema of Abstraction (or Specification or Comprehension). The Set Builder Axiom Schema.
And a number of other axioms
(x=y) ≡∀z(zεx≡zεy)) Axiom of Extensionality