Intersection of Power Sets

Theorem

The intersection of the power sets of two sets $S$ and $T$ is equal to the power set of their intersection:

$\powerset S \cap \powerset T = \powerset {S \cap T}$

$\ds X$

$\in$

$\ds \powerset {S \cap T}$

$\ds \leadstoandfrom \ \ $

$\ds X$

$\subseteq$

$\ds S \cap T$

Definition of Power Set

$\ds \leadstoandfrom \ \ $

$\ds X$

$\subseteq$

$\ds S \land X \subseteq T$

Definition of Set Intersection

$\ds \leadstoandfrom \ \ $

$\ds X$

$\in$

$\ds \powerset S \land X \in \powerset T$

Definition of Power Set

$\ds \leadstoandfrom \ \ $

$\ds X$

$\in$

$\ds \powerset S \cap \powerset T$

Definition of Set Intersection

$\blacksquare$

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: Exercise $5 \ \text{(a)}$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $7 \ \text{(ii)}$