# Axiom:Axiom of Pairing

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## 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**.

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.

## Also see

- Equivalence of Definitions of Axiom of Pairing: Both forms of the axiom are equivalent, assuming the Axiom of Subsets.
- Definition:Ordered Pair
- 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.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 3$: Unordered Pairs - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets

- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html