Zermelo-Fraenkel axioms

The Zermelo-Fraenkel axioms form one of several possible formulations of axiomatic set theory.

The axioms
There are eight Zermelo-Fraenkel (ZF) axioms:
 * 1) Axiom of extensionality : If X and Y have the same elements, then X=Y
 * 2) Axiom of pairing : For any a and b there exists a set {a, b} that contains exactly a and b
 * 3) Axiom schema of separation : If &phi; is a property with parameter p, then for any X and p there exists a set Y that contains all those elements u&isin;X that have the property &phi;; that is, the set Y={u&isin;X|&phi;(u, p)}
 * 4) Axiom of union : For any set X there exists a set Y=&cup;X, the union of all elements of X
 * 5) Axiom of power set : For any X there exists a set Y=P(X), the set of all subsets of X
 * 6) Axiom of infinity : There exists an infinite set
 * 7) Axiom schema of replacement : If f is a function, then for any X there exists a set Y, denoted F(X) such that F(X)={f(x)|x&isin;X}
 * 8) Axiom of regularity : Every nonempty set has an &isin;-minimal element

If to these is added the axiom of choice, the theory is designated as the ZFC theory:

&emsp;9. Axiom of choice : Every family of nonempty sets has a choice function