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https://en.wikipedia.org/wiki/Binary_relation
4 Sets versus classes
Sets versus classes
Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory. For example, if we try to model the general concept of "equality" as a binary relation =, we must take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.

In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set A, that contains all the objects of interest, and work with the restriction =A instead of =. Similarly, the "subset of" relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set of a specific set A): the resulting set relation can be denoted by ⊆A. Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown that assuming ∈ to be defined over all sets leads to a contradiction in naive set theory.

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