What role does the axiom of pairing play in building sets?

In set theory, the axiom of pairing is a fundamental principle used to construct new sets from existing ones. It serves as a necessary tool for compounding sets, ensuring that if you have two existing sets, you can form a new set that contains exactly those two elements.1

While the axiom schema of separation allows for the construction of subsets, the axiom of pairing—along with the axiom of union and the axiom of power set—provides the essential means to build more complex structures from simpler ones. By enabling the creation of these basic pairings, the axiom facilitates the definition of various mathematical concepts, including relations, functions, and Cartesian products, which are foundational to the development of mathematics.1

It is worth noting that in some advanced axiomatic systems, the axiom of pairing can be deduced when applying the axiom schema of replacement in conjunction with the axiom of power set.2