There's no set $X$ such that $P(X)\subset X$.
47697
Russell's paradox: a set cannot contain its own powerset
132382
Showing a set is nowhere dense in $C([0,1])$
30660
Empty set does not belong to empty set
109054
Given an infinite set of cardinality $X$, is there a chain in $\mathcal P(X)$ of the same cardinalty of $\mathcal P(X)$?
93771
Prove that for any set $A$ there is some $x \notin A$
14496
Is there any example of a non-measurable set whose proof of existence doesn't appeal to the Axiom of choice?
136106
Is every open subset of $ \mathbb{R} $ uncountable?
99064
Uncountable set with exactly one limit point
93750
Can a set be neither open nor closed?
122762
Proving that the limit doesn't exist for $x$ other than $0$
16056
Is empty set element of every set if it is subset of every set?
98580
Lebesgue-measurable sets requiring the Axiom of Choice to construct
71653
Why is the Cartesian product of a set $A$ and empty set an empty set?
136755
Direct proof of empty set being subset of every set
116527
Validating a proof for necessary and sufficient conditions on sets A & B such that $P(A) \cup P(B) = P(A\cup B)$
130771
Sets and the principle of inclusion and exclusion
64075
Prove that a set of connectives is inadequate