Proof — Axiom Schema of Separation
Statement
For any set x and any formula φ(u, p), there exists a set y = {u ∈ x : φ(u, p)} containing exactly those elements of x that satisfy φ. Also called the Axiom of Subsets or Comprehension. (Schema: one axiom for each formula φ.)
Formula
∀x ∀p ∃y ∀u (u ∈ y ↔ u ∈ x ∧ φ(u, p))
Source
Zermelo proposed 1908 (theorem); Fraenkel established ZF 1922 (axiom).
Domains
mathematics