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Let $V$ be a set and let $P$ be a property, such that for no $F\subseteq V, \ P(F)$ is true.

Is then $\left\{ F\subseteq V | P(F) \right\} = \emptyset$ or $= \left\{ \emptyset \right\} $ ?

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If $\emptyset$ is in the set then $P(\emptyset)$ must be true. – François G. Dorais Jul 14 '11 at 15:07
up vote 6 down vote accepted

If $A=\{F\subseteq V\mid P(F)\}=\{\varnothing\}$ then it means that $\varnothing\in A$, therefore $P(\varnothing)$ is true.

If no set is such that the property holds, then $A=\varnothing$, i.e. it has no elements.

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