Existence of families of sets whose elements are incomparable in terms of $\in$ There exist families of sets whose elements are comparable in terms of $\in$, like for example the set of finite von Neumann ordinals, there exist such that their elements are incomparable in terms of $\in$ (in the sense that for no two elements $x$ and $y$: $x\in y$ or $y\in x$), like for example $\\{\\{1\\},\\{2\\},\\{3\\},\ldots\\}$. My question is: can we construct for an arbitrary cardinality $\kappa$, a family $\mathcal{A}$ of sets of cardinality $\kappa$, whose elements are pairwise incomparable in terms of $\in$: \\[ (\forall {x,y\in\mathcal{A}})\,(x\notin y\wedge y\notin x)\ ? \\]

If you take any set of size $\kappa$, $X$, that none of its elements are singletons, then $\\{\\{x\\}\mid x\in X\\}$ is such set. For example, take $X=\kappa\setminus\\{1\\}$.