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In my experience in analysis, basically the only place where it is actually important to distinguish sets from proper classes arises when one wishes to invoke Zorn's lemma to locate a maximal object in some non-empty partially ordered set $X$ in which all chains are bounded (e.g. to create a maximal hyperplane, a maximal filter, a maximally defined bounded linear functional, etc.). Here it is crucial that $X$ is "small" enough to be an actual set (e.g. it is a collection of subsets of some space $V$ that is already known to be a set, or a set of functions from $V$ to yet another set). For instance, one cannot use Zorn's lemma to construct a maximal set in the class of all sets, or a maximal group in the class of all groups, or a maximal vector space in the class of all vector spaces. (Such maximal objects, if they existed, would soon lead to contradictions of the flavour of Russell's paradox or the Burali-Forti paradox; not coincidentally, one of the standard proofs of Zorn's lemma proceeds by contradiction, using the axiom of choice to embed all the ordinals into $X$, which can then be used to set up the Burali-Forti paradox.)