I've been given a task that reads:
Prove that given formulae is correct with the use of set theory axioms: $(\forall a)(\exists b)(\forall c)((c \in b) \iff (\exists d \in a)(c \subset d))$
Now, how I interpret it: For every set a there exists a set b, which is a power set of some element from set a. Some friends have suggested another possible interpretation, that would expect b to be a set of power sets of all emelements from a.
I want to ask what's the correct interpretation here, why is the other one wrong and if there are some rules according to which I'd be able to decide what interpretation is right if I'm presented with similar task in the future.
The rest of the task is fairly straightforward, I use axiom of power sets to prove the existence of the power set and then something like axiom of existence which proves that every set has at least one element in it. The main problem lies in the interpretation.