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:

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

Proof

\(\ds s\)

\(\in\)

\(\ds \set {x \in S: \map P x}\)

\(\ds \leadsto \ \ \)

\(\ds s\)

\(\in\)

\(\ds \set {x \in S \land \map P x}\)

\(\ds \leadsto \ \ \)

\(\ds s\)

\(\in\)

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

Definition of Set Definition by Predicate

\(\ds \leadsto \ \ \)

\(\ds s\)

\(\in\)

\(\ds \set {x \in S}\)

Rule of Simplification

\(\ds \leadsto \ \ \)

\(\ds s\)

\(\in\)

\(\ds S\)

Definition of Element

\(\ds \leadsto \ \ \)

\(\ds \set {x \in S: \map P x}\)

\(\subseteq\)

\(\ds S\)

Definition of Subset

$\blacksquare$

Sources

• 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