Is it possible to prove that equal power sets imply equal sets using the ZF axioms?

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- Jun 29th 2012, 08:56 AMcpcookEqual powers sets in ZF theory
Is it possible to prove that equal power sets imply equal sets using the ZF axioms?

- Jun 29th 2012, 09:05 AMPlatoRe: Equal powers sets in ZF theory
- Jun 29th 2012, 10:51 AMcpcookRe: Equal powers sets in ZF theory
I cannot use the subset notation. We can only use the "element of". That is our only operator in the universe we are using.

- Jun 29th 2012, 11:09 AMPlatoRe: Equal powers sets in ZF theory
- Jun 29th 2012, 02:28 PMemakarovRe: Equal powers sets in ZF theory
Well, yes, Zermelo–Fraenkel set theory has only two predicates: equality and membership. However, $\displaystyle A\subseteq B$ is easily expressible as $\displaystyle \forall x.\,x\in A\to x\in B$. Subset relation is used in the axiom of power set.

- Jun 29th 2012, 03:45 PMDevenoRe: Equal powers sets in ZF theory
in this case, i would be tempted to prove the contrapositive: distinct sets give rise to distinct power sets. you can use the same argument as in Plato's post:

if x is in A\B, then {x} is in P(A)\P(B).