# Axiom:Axiom of Pairing/Set Theory

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## 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

- Equivalence of Definitions of Axiom of Pairing: Both forms of the axiom are equivalent, assuming the axiom of specification.
- Definition:Doubleton

### 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.