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

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 $$Y:=\{z\in X\mid p(z,X_1,\ldots,X_n)\}.$$

| | | | | Contributors: *bookofproofs* | References: [656], [983]

## 1.**Proof**: *(related to "Justification of the Set-Builder Notation")*

[983] **Ebbinghaus, H.-D.**: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3

[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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