[FOM] A question about a new definition of Cardinality.

[Zuhair Abdul Ghafoor Al-Johar]

Dear FOMers,

Cardinality is a one place function satisfying the following
basic property.

for all x, for all y: Cardinality(x) = Cardinality(y) iff  x bijective to y

Now Let C(x) be the set of all sets hereditarily subnumerous to
P(P(x)) that are equinumerous to x.

Is there a known model of ZF in which C does not satisfy the above
basic property?

Best Regards

Zuhair