In Moschovakis Notes on set theory book there is an exercise that says
Prove that $A$ is a set if and only if $A$ is a member of a class.
(Here a class is either a set or a unary definite condition.) One direction of the proof is trivial because the "set of all sets" is a class. To prove the other direction I used, literally, the axiom of pairing stated in the book:
Axiom of pairing: for any two objects $x$ and $y$ there is a set who only elements are $x$ and $y$.
Then an element $x$ of a class is an object (of the universe of the theory), and consequently $\{x\}$ is a set. Now using the axiom of union we find that $\bigcup\{x\}=x$ is a set.
QUESTIONS:
It is my proof correct?
I find unsatisfying the use of a particular form of the axiom of pairing when, in general, it is not stated for objects if not just for sets. Then, supposing that the above proof is correct, there is som other way to prove the statement?