Full Idea
Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}.
Gist of Idea
The singleton is defined using the pairing axiom (as {x,x})
Source
Herbert B. Enderton (Elements of Set Theory [1977], 2:19)
Book Reference
Enderton,Herbert B.: 'Elements of Set Theory' [Posts + Telecoms 2006], p.19
A Reaction
An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter!