Axiom:Axiom of Pairing

Axiom
For any two sets, there exists a set to which only those two 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 sets that you have already created.

Otherwise known as the Axiom of the Unordered Pair.

This can be deduced from the Axiom of Infinity and the Axiom of Replacement.

It can also be deduced from the Axiom of Powers and the Axiom of Replacement

In both cases the set $2 = \left\{\varnothing,\left\{\varnothing\right\}\right\}$ is used with the axiom of replacement as the domain for a function whose range is $\left\{A, B \right\}$

A suitable function would be $ \left( y = \varnothing \land z = A \right) \lor \left( y = \left\{{ \varnothing }\right\} \land z = B \right) $

The set $2$ is shown to exist either as the set of all subsets of the set of all subsets of the empty set, or as a member of the infinite set whose existence is asserted by the axiom of infinity.

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. Both forms of the axiom are equivalent, assuming the Axiom of Subsets.