In set theory and its applications throughout mathematics, a subclass is a class contained in some other class in the same way that a subset is a set contained in some other set.
That is, given classes A and B, A is a subclass of B if and only if every member of A is also a member of B. If A and B are sets, then of course A is also a subset of B. In fact, it's enough that B be a set; the axiom of specification essentially says that A must then also be a set.
As with subsets, the empty set is a subclass of every class, and any class is a subclass of itself. But additionally, every class is a subclass of the class of all sets. Accordingly, the subclass relation makes the collection of all classes into a Boolean lattice, which the subset relation does not do for the collection of all sets. Instead, the collection of all sets is an ideal[?] in the collection of all classes. (Of course, the collection of all classes is something larger than even a class!)
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