Showing a class is a set.

Question

The problem is to determine that a given class is or is not a set only using the basic axioms of modern set theory and some examples of classes that are or are not sets. The class in question is $$\mathfrak{A} = \{\mathcal{P}(X) | X \subseteq A \},$$ where $A$ is an arbitrary set. Here is my attempt at the problem:

First prove a lemma,

---

Lemma: Let $A$ be a set. Then $$\bigcup_{X \subseteq A} \mathcal{P}(X) \subseteq \mathcal{P}(\bigcup_{X \subseteq A} X)$$ (pf): Let $$S \in \bigcup_{X \subseteq A} \mathcal{P}(X)$$ $\Rightarrow$ $S \subseteq X$ for at least one $X \subseteq A$

$\Rightarrow$ $S \subseteq \bigcup_{X \subseteq A} X$

$\Rightarrow$ $S \in \mathcal{P}(\bigcup_{X \subseteq A} X)$

$\therefore \bigcup_{X \subseteq A} \mathcal{P}(X) \subseteq \mathcal{P}(\bigcup_{X \subseteq A} X)$

---

Now by the unionset axiom $$\bigcup \mathfrak{A} = \bigcup \mathcal{P}(X) \subseteq \mathcal{P}(\bigcup X) \subseteq \mathcal{P}(A).$$

Then by a previous exercise $\bigcup \mathfrak{A} \subseteq \mathcal{P}(A) \Leftrightarrow \bigcup \mathfrak{A}$ is a set as $\mathcal{P}(A)$ is a set. Therefore, $\mathfrak{A}$ is a set as $\bigcup \mathfrak{A}$ is a set.

Quod Erat Demonstratum?

Answer 1

It's easiest to use the replacement schema. $\varphi(a,b)$ being $b=\mathcal P(a)$.

Your proof is somewhat unclear. The lemma is a good start, but you need to argue now that $\frak A$ is a subset of $\mathcal{P(P(}A))$.

Answer 2

It is much easier to simply define $\mathfrak U$ as a subset of $\mathscr P\mathscr P A$.