The “difficulty” is now known as Russell’s paradox. As Frege knew, a membership predicate “x ∈ y” can be defined as “∃F(y = {u | Fu} ∧ Fx).” That is, x is a member of y when y is the extension of a concept F under which x falls. Consider now the Russell class r, defined as {x | x ∉ x}. Is r a member of itself?
