# Axiom of Pairing from Axiom of Subsets

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## Theorem

Let it be supposed that there exists a set which contains at least two elements.

Then the Axiom of Pairing is a consequence of the Axiom of Specification.

## Proof

Let $A$ be a set which contains at least two elements.

Let $a$ and $b$ be any two elements of $A$.

Let $\map P x$ be the propositional function:

- $\map P x := \paren {x = a \lor x = b}$

Then we may use the Axiom of Specification to define $B$ as:

- $x \in B \iff \set {x \in A: \map P x}$

Hence we can define:

- $B := \set {a, b}$

for any two $a, b \in A$.

$\blacksquare$

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 3$: Unordered Pairs