There Exists No Universal Set/Proof 2

Theorem

There exists no set which is an absolutely universal set.

That is:

$\map \neg {\exists \, \UU: \forall T: T \in \UU}$

where $T$ is any arbitrary object at all.

That is, a set that contains everything cannot exist.

Proof

Aiming for a contradiction, suppose such a $\UU$ exists.

By No Injection from Power Set to Set, $\powerset \UU$ has no injection to $\UU$.

Let $f : \powerset \UU \to \UU$ be the identity mapping.

By Identity Mapping is Injection, it is an injection, which leads to a contradiction.

$\blacksquare$