# Axiom:Axiom of Pairing/Set Theory

## Axiom

### Formulation 1

For any two sets, there exists a set to which only those two sets are elements:

$\forall A: \forall B: \exists x: \forall y: \paren {y \in x \iff y = A \lor y = B}$

Thus it is possible to create a set that contains as elements any two sets that have already been created.

### Formulation 2

For any two sets, there exists a set containing those two sets as elements:

$\forall A: \forall B: \exists x: \forall y: \paren {y \in x \implies y = A \lor y = B}$

Thus it is possible to create a set that contains as elements at least two sets that have already been created.

## Also known as

The axiom of pairing is also known as the axiom of the unordered pair.

Some sources call it the pairing axiom.

## Also see

### Relation to other axioms

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.