Proof of Elements Are Sets, and the Empty Set and the Pair of Two Sets Exist and Are Unique

Clause elements. Let $X$ be an element of a class $Y$, that is, $X\in Y$. Then there is a class of which $X$ is an element, so $X$ is a set by Classes, Membership, Sets and Proper Classes §set.

Clause empty. By Axiom of the Empty Set there is a set $e$ such that no set $u$ satisfies $u\in e$. In fact $e$ has no element at all: if a class $w$ satisfied $w\in e$, then $w$ would be a set by Classes, Membership, Sets and Proper Classes §set, contrary to the choice of $e$. Now let $C$ be any class with no element. For every set $z$, both $z\in C$ and $z\in e$ are false, so $z\in C$ holds if and only if $z\in e$ holds; by Axiom of Extensionality for Classes, $C=e$. Thus every class with no element equals $e$; in particular every set with no element equals $e$, so $e$ is the only set with no element.

Clause pair. Let $x$ and $y$ be sets. By Axiom of Pairing there is a set $z$ such that, for every set $u$, $u\in z$ if and only if $u=x$ or $u=y$. If $z'$ is a set with the same property, then for every set $u$ we have $u\in z$ if and only if ($u=x$ or $u=y$) if and only if $u\in z'$, so $z=z'$ by Axiom of Extensionality for Classes. Hence $z$ is unique.