I wonder how to proof that: "$B := \{ P(E) | E \in A \}$", using the Zermelo-Fraenkel axioms and given the information in the Headline.
For example:
"$A = \{\{a\},\{b,c\}\}$", thus "$B = \{ \{ \{ \emptyset, \{ a \} \}, \{ \emptyset, \{ b \}, \{ c \}, \{ b, c \} \} \}$".
Using the axiom of power sets: "$\forall x | x \in P \Leftrightarrow x \subseteq E $" with "$P$" being "$P(E)$" as well as "$E \in A$", how do I proof the existence of "$B$"?
I understand that using the axiom of power sets we suppose the existence of "$P(E)$" for any "$E \in A$". How can I now use atleast one of the Zermelo-Fraenkel axioms to show that "$B$", containing every "$P(E)$" must exist? Does "$A$" simply containing more than one element force the collection of every "$P(E)$" to exist in a separate set? I guess no, but I can not explain. Can someone please help me to understand which axiom should be used and why?
Thank you so much in advance, I am looking forward to your answer!