Subset of Set with Propositional Function

Theorem

Let $S$ be a set.

Let $P: S \to \set {\T, \F}$ be a propositional function on $S$.

Then:

$\set {x \in S: \map P x} \subseteq S$

$\ds s$

$\in$

$\ds \set {x \in S: \map P x}$

$\ds \leadsto \ \ $

$\ds s$

$\in$

$\ds \set {x \in S \land \map P x}$

$\ds \leadsto \ \ $

$\ds s$

$\in$

$\ds \set {x \in S} \land \map P s$

Definition of Set Definition by Predicate

$\ds \leadsto \ \ $

$\ds s$

$\in$

$\ds \set {x \in S}$

$\ds \leadsto \ \ $

$\ds s$

$\in$

$\ds S$

Definition of Element

$\ds \leadsto \ \ $

$\ds \set {x \in S: \map P x}$

$\subseteq$

$\ds S$

Definition of Subset

$\blacksquare$

1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 4$
1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology