[University Set Theory] Choice of axioms

This is not actually a homework question, just a question I came up with on my own while reading through a set theory textbook.

Why use the pairing axiom and not an axiom defining a singleton as a basis for set theory? It seems to me like an arbitrary added bit of complexity. The textbook I am reading writes the axiom as following: ∀u ∀v ∃B ∀x(x∈B ⇔ x=u or x=v).

What is the justification for not using the following, or something similar?: ∀u ∃B ∀x(x∈B ⇔ x=u)

Combining this with the "Union Axiom" [∀a ∀b ∃B ∀x(x∈B ⇔ x∈a or x∈b)]it is still easy to form all finite sets (including pairs), and this eliminates the added logical step as defining a singleton as a pair in which both members are the same.