Let me try the first one:
(to answer the question you only need the first equation)
Using the Bounded Comprehension Principle, show that if x and y are sets then
I'm just not sure the Bounded Comprehension Principle applies here... could someone explain this?
P(x) is the Power Set of x, that is .
Since x is a set, so is P(x) (by the power set axiom).
I realize that you need to do all of them. I did the first one to start you off. If you understand this one, then you should at least be able to attempt the other two. Give them a try, show your thoughts, and I'll help out if you get stuck.
I've never heard that definition, but it is equivalent to the one I have given you. For example, if has rank , the rank of can be expressed in terms of (I think it has rank ). Conversely you can find a bounding set for a collection of sets of rank less than a fixed ordinal.