Equivalence of Formulations of Axiom of Empty Set for Classes

Theorem
In the context of class theory, the following formulations of the Axiom of the Empty Set are equivalent:

Proof
It is assumed throughout that the axiom of extensionality and the axiom of specification both hold.

Formulation $1$ implies Formulation $2$
Let formulation $1$ be axiomatic:

There exists a set that has no elements:

From the Axiom of Specification it follows that the $x$ so created is a class, and we label it $\O$.

But this $\O$ is a set.

Hence it follows that the truth of formulation $2$ follows from acceptance of the truth of formulation $1$.

Formulation $2$ implies Formulation $1$
Let formulation $2$ be axiomatic:

By the Axiom of Specification, we can define the class $\O$ as:


 * $\O := \set {y: \lnot {y \ne y} }$

which is a class with no elements.

We have that $\O$ is a set.

By the Axiom of Extensionality we have that $\O$ is unique.

From Equivalence of Formulations of Axiom of Empty Set, we can express $\O$ as:
 * $\O := \set {y: \lnot {y \in \O} }$

That is, the truth of formulation $1$ follows from acceptance of the truth of formulation $2$.