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if f maps x onto P(x) then P(x) contains at least one fuzzy element, ie, x contains at least one fuzzy subset.

the contrapositive of this is that if x does not contain at least one fuzzy subset then there is no map from x onto P(x).

U=P(U) does contain a fuzzy element, namely the S in russell's 'paradox'.

a set is called fuzzy if there is a set whose membership value in that set is neither true nor false. otherwise the set is called crisp.

if you want version 10, then use the same link but replace the 9 with a 10.