Full Idea
Axiom of Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z). Any pair of entities must form a set.
Gist of Idea
Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z)
Source
Kenneth Kunen (Set Theory [1980], §1.6)
Book Reference
Kunen,Kenneth: 'Set Theory: Introduction to Independence Proofs' [North-Holland 1980], p.12
A Reaction
Repeated applications of this can build the hierarchy of sets.