Proposition: All sets obtained from successive applications of the axiom of pairing to the empty set are unique.
My attempts on this so far are to start with the empty set and pair upwards to an arbitrary number of nested sets, but my professor slams this approach because we haven't yet constructed the idea of "numbers", so this proof is meaningless. Instead I'm supposed to "start from above" with "any number(!!) of applications of pairing, and then start stripping away using extension."
What the hell is going on? How do I prove this?