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I know that in axiomatized set theory the empty set is disjoint with itself. Because it has no members, the empty set cannot have any members in common with itself. This is common sense in set theory.

But if we say that set A is disjoint with set B, aren't we then implying that A is not identical with B? But how can the empty set be non-identical with itself? To me this has paradox written all over it, but to me surprise all textbooks about set theory treat the disjointness of the empty set with itself as something trivial and not very remarkable. Please let me know what you think!