Another way is to work in a one-sorted theory with a unary relation used to distinguish sets and urelements.
[2] Thus, standard expositions of the canonical axiomatic set theories ZF and ZFC do not mention urelements (for an exception, see Suppes[3]).
Adding urelements to the system New Foundations (NF) to produce NFU has surprising consequences.
Holmes (1998) takes these facts as evidence that NFU is a more successful foundation for mathematics than NF.
[6] In finitist set theory, urelements are mapped to the lowest-level components of the target phenomenon, such as atomic constituents of a physical object or members of an organisation.
ZF set theory with the axiom of regularity removed cannot prove that any non-well-founded sets exist (unless it is inconsistent, in which case it will prove any arbitrary statement), but it is compatible with the existence of Quine atoms.
Other non-well-founded theories may admit many distinct Quine atoms; at the opposite end of the spectrum lies Boffa's axiom of superuniversality, which implies that the distinct Quine atoms form a proper class.