Axiom of Pairing from Powers and Replacement

Theorem
The Axiom of Pairing is a consequence of:
 * the Axiom of Powers

and
 * the Axiom of Replacement.

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

A suitable mapping would be:
 * $\left({y = \varnothing \land z = A}\right) \lor \left({y = \left\{{\varnothing}\right\} \land z = B}\right)$

The set $2$ is shown to exist as the set of all subsets of the set of all subsets of the empty set.