Working in $\text{NFU}$, let $Ur$ be the set of all empty objects except a specific empty object $\emptyset$ that stands as the empty set, let $Set$ be the set of all non empty objects and $\emptyset$. Now let $P(x)$ denote the set of all subsets of $x$, i.e.

$$ P(x)=\{y \ | \ y \in Set \wedge \forall z \in y \ (z \in x)\}$$

Now lets add the axiom that $|Ur|>|Set|$.

It's clear that we have: $|Set|=|P(Ur)|$

Where cardinality $``||"$ is defined after Frege.

What is the proof that $|P(Set)|<|P(Ur)|$?


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