Some more thoughts on OP.

A set is defined by its contents, described or listed.

Named in broadest sense: denote, designate, identify, refer to, etc.

A common way of naming a set is to use {}. “{a,b,c}” is the name of {a,b,c}.

A is a subset of B if A only contains elements of B doesn’t work for {}, except in the negative sense, and doesn’t prove existence.

A is a subset of B if you can get A by removing none or more elements of B establishes existence of {}, gives that all {} are identical, and shows the empty set is a subset of every set.