Axiom of Pairing

For any sets ‘A’ and ‘B’ there exists a set C={A,B} that contains exactly ‘A’ and ‘B’.

For the special case where A=B, C={A}, therefore, {A} exists.

Appendix A:

It is intriguing, why did we have a need to postulate the existence of a set with two sets?

Well, we may have found later why this postulate must be postulated: when working with Axiom of Foundation we need the existence of nonempty set A to imply the existence of {A}.

Since A is nonempty, we’re going to write A = {a} with element ‘a’ being whatever it is. The existence of this A means {{a}} exists, because {A}={{a}}.