Urelements

axiom of Zermelo–Fraenkel set theory asserting that set equality is determined by the membership relation

In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Ur-elements are also not identical with the empty set.

axiomatic set theory permitting set comprehension by stratified formulae, hence with a universal set, but in which the singleton map 𝑥↦{𝑥} fails to exist