Sets question, without Zorn's lemma Is there any proof to $|P(A)|=|P(B)| \Longrightarrow |A|=|B|$ that doesn't rely on Zorn's lemma (which means, without using the fact that $|A|\neq|B| \Longrightarrow |A|<|B|$ or $|A|>|B|$ ) ?
Thank you!

Even with Zorn's Lemma, one cannot (under the usual assumption that ZF is consistent) prove that if two power sets have the same cardinality, then the sets have the same cardinality. 
