# Axiom of Pairing from Infinity and Replacement

## Theorem

The Axiom of Pairing is a consequence of:

the Axiom of Infinity

and

the Axiom of Replacement.

## Proof

The set $2 = \set {\O, \set \O}$ is used with the Axiom of Replacement as the domain for a mapping whose image is $\set {A, B}$.

A suitable mapping would be:

$\paren {y = \O \land z = A} \lor \paren {y = \set \O \land z = B}$

The set $2$ is shown to exist as a member of the infinite set whose existence is asserted by the axiom of infinity.

$\blacksquare$