# Axiom:Axiom of Pairing/Set Theory

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## Axiom

### Strong Form

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

- $\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \iff z \in c}$

### Weak Form

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

- $\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \implies z \in c}$

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

- Results about
**Axiom of Pairing**can be found**here**.

### 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.