Equivalence of Definitions of Axiom of Pairing for Classes

Theorem
The following formulations of the axiom of pairing in the context of class theory are equivalent:

$(1)$ implies $(2)$
Let formulation $1$ of the axiom of pairing be assumed:


 * The class $\set {a, b}$ is a set.

Then we have that $c = \set {a, b}$ is a set such that both $a \in c$ and $b \in c$.

Thus formulation $2$ of the axiom of pairing is seen to hold.

$(2)$ implies $(1)$
Let formulation $2$ of the axiom of pairing be assumed:


 * There exists a set $c$ such that $a \in c$ and $b \in c$.

Then the class $\set {a, b}$ is a subclass of $c$.

By the axiom of swelledness, every subclass of $c$ is a set.

That is, $\set {a, b}$ is aset.

Thus formulation $1$ of the axiom of pairing is seen to hold.