Singleton of Subset is Element of Powerset of Powerset

Theorem

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

Then:

$\set S \in \powerset {\powerset T}$

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

Proof

\(\ds S\)

\(\subseteq\)

\(\ds T\)

\(\ds \leadsto \ \ \)

\(\ds S\)

\(\in\)

\(\ds \powerset T\)

Definition of Power Set

\(\ds \leadsto \ \ \)

\(\ds \set S\)

\(\in\)

\(\ds \powerset {\powerset T}\)

Element in Set iff Singleton in Powerset

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 2$. Sets of sets