# Axiom:Axiom of Pairing/Set Theory

## 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

• 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.