Axiom of Pairing from Axiom of Specification

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 Subsets.

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

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

Let $P \left({x}\right)$ be the propositional function:
 * $P \left({x}\right) := \left({x = a \lor x = b}\right)$

Then we may use the Axiom of Subsets to define $B$ as:
 * $x \in B \iff \left\{{x \in A: P \left({x}\right)}\right\}$

Hence we can define:
 * $B := \left\{{a, b}\right\}$

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