The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory...

\lnot (u\in u)\}.} Thus, the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique...

implies both propositional and functional extensionality. Extensionality principles are usually assumed as axioms, especially in type theories where computational...

Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity...

, the power set of x {\displaystyle x} , consisting precisely of the subsets of x {\displaystyle x} . By the axiom of extensionality, the set P ( x )...

assume any axioms except the axiom of extensionality and the axiom of induction—a natural number is either zero or a successor and each of its elements...

(See the Lévy hierarchy.) Axiom of extensionality: Two sets are the same if and only if they have the same elements. Axiom of induction: φ(a) being a formula...

axiom of extensionality to show that this set C is unique. We call the set C the pair of A and B, and denote it {A,B}. Thus the essence of the axiom is:...

{\displaystyle S} of A whose members are precisely the members of A that satisfy φ ( x ) {\displaystyle \varphi (x)} . By the axiom of extensionality this set...

B)\to A=B.} This axiom is identical to the axiom of extensionality found in many other set theories, including ZF. Any element or a subset of a set is a set...