Equivalence of Formulations of Axiom of Pairing for Classes

Theorem
The following formulations of the Axiom of Pairing in the context of class theory are equivalent:

Proof
It is assumed that all classes are subclasses of a basic universe $V$.

$(1)$ implies $(2)$
Let formulation $1$ of the Axiom of Pairing be assumed:

Thus 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:

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

We have that $c$ is a subclass of a basic universe $V$.

Hence by the Axiom of Swelledness, every subclass of $c$ is a set.

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

Thus formulation $1$ of the Axiom of Pairing is seen to hold.