Powerset is not Subset of its Set

Theorem

Let $A$ be a set.

Then:

$\powerset A \not \subseteq A$

Proof 1

Aiming for a contradiction, suppose that $\powerset A \subseteq A$, and define:

$C = \set {x \in \powerset A : x \notin x}$

We have that $C \subseteq \powerset A$, as it contains only the $x \in \powerset A$ meeting the condition $x \notin x$.

Since $\powerset A \subseteq A$, we have:

$C \subseteq A$

and thus

$C \in \powerset A$

We can derive a similar contradiction to Russell's Paradox.

If $C \in C$, then it must meet $C$'s condition that $C \notin C$.

If $C \notin C$, then it meets $C$'s condition for $C \in C$.

$\blacksquare$

Proof 2

Aiming for a contradiction, suppose that $\powerset A \subseteq A$.

Let $I: \powerset A \to A$ be the identity mapping.

$I$ is an injection by Identity Mapping is Injection.

But by No Injection from Power Set to Set, this is a contradiction.

$\blacksquare$

Proof 3

Aiming for a contradiction, suppose that $\powerset A \subseteq A$.

Since $A \in \powerset A$, this implies:

$A \in A$

But this contradicts Set is Not Element of Itself.

$\blacksquare$

Also see

• No Injection from Power Set to Set