Axiom:Axiom of Pairing

Axiom
For any two sets, there exists a set to which only those two [Definition:Set|sets]] are elements:


 * $\forall A: \forall B: \exists x: \forall y: \left({y \in x \iff y = A \lor y = B}\right)$

Thus it is possible to create a set containing any two [Definition:Set|sets]] that you have already created.

Otherwise known as the Axiom of the Unordered Pair.

The Axiom of Pairing can alternatively be stated as:


 * $\forall A: \forall B: \exists x: \forall y: \left({y \in x \implies y = A \lor y = B}\right)$

which guarantees the existence of a set that contains at least two elements.

Also see
The Axiom of Pairing can be deduced as a consequence of:


 * $(1): \quad$ The Axiom of Infinity and the Axiom of Replacement: see Axiom of Pairing from Infinity and Replacement


 * $(2): \quad$ The Axiom of Powers and the Axiom of Replacement: see Axiom of Pairing from Powers and Replacement.


 * Equivalence of Definitions of Axiom of Pairing: Both forms of the axiom are equivalent, assuming the Axiom of Subsets.