Please recall the different basic possibilities to describe sets, in which the set-builder notation used the curly brackets, inside which we describe the definite properties of the set elements. The following corollary to the axiom of separation allows justification of this notation.# Corollary: Justification of the Set-Builder Notation
(related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
Let $p(z,X_1,\ldots,X_n)$ be an atomic formula in predicate logic, in which the $z$ is a free variable and in which $x_1,\ldots,x_n, X$ are sets. Let $Y$ be given, which fulfills the property of the axiom of separation, i.e. $$\forall x_1,\ldots,x_n \forall X~\exists Y~\forall z~(z\in Y \Leftrightarrow z\in X\wedge p(z,x_1,\ldots,x_n)).$$ Then $Y$ is unique. Therefore, we can therefore use the formula to define the set $Y,$, justifying the set-builder notation $$Y:=\{z\in X\mid p(z,x_1,\ldots,x_n)\}.$$
Proofs: 1
Explanations: 1