Power Set of Subset

Theorem

Let $S \subseteq T$ where $S$ and $T$ are both sets.

Then:

$\powerset S \subseteq \powerset T$

where $\powerset S$ denotes the power set of $S$.

$\ds X$

$\in$

$\ds \powerset S$

$\ds \leadsto \ \ $

$\ds X$

$\subseteq$

$\ds S$

Definition of Power Set

$\ds \leadsto \ \ $

$\ds X$

$\subseteq$

$\ds T$

$\ds X$

$\in$

$\ds \powerset T$

Definition of Power Set

$\blacksquare$

1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers
1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 2$. Sets of sets
1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets: Exercise $1 \ \text{(g)}$
2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 6$ The power axiom: Exercise $6.1. \ \text {(c)}$