structural:expansion, stylistic:tone
Axiomatic definition

Ordered pairs can also be introduced in Zermelo–Fraenkel set theory (ZF) axiomatically by just adding to ZF a new function symbol f {\displaystyle f} of arity 2 (it is usually omitted) and a defining axiom for f {\displaystyle f}: f (a 1, b 1) = f (a 2, b 2) if and only if a 1 = a 2 and b 1 = b 2. {\displaystyle f(a_{1},b_{1})=f(a_{2},b_{2}){\text{ if and only if }}a_{1}=a_{2}{\text{ and }}b_{1}=b_{2}.}

This definition is acceptable because this extension of ZF is a conservative extension.

The definition helps to avoid so called accidental theorems like (a,a) = {{a}}, and {a} ∈ (a,b), if Kuratowski's definition (a,b) = {{a}, {a,b}} was used.