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*Proposition:*
Every set is smaller than its powerset:

forallS:Sis smaller thanP(S).

*Proof:*
Take arbitrary `S`. `S` is not larger than **P**(`S`)
because we can define

Assume that

f:S->^{injective}P(S)f(x) := {x}.

Take `A` := {`x` in `S`: `x` not in `f`(`x`)}. Since `f` is surjective and `A` subset `S`, i.e., `A` in **P**(`S`),
we have some `a` in `S` with
`f`(`a`) = `A`. But then we know
`a` in `A` <=> `a` not in `f`(`a`)
<=> `a` not in `A`.

Author: Wolfgang Schreiner

Last Modification: December 7, 1999