I think that "u+v is in subspace S" is equivalent to "u+v is in S, which is a subspace". Then this is written as .
Emakarov is correct. In addition, I would say that the notation means that is one object, and it is a member of the set The second notation you mentioned is (I think) and this means that is a set, and every element in is also in So, corresponds to one element being in a set, whereas corresponds to what's on the left being a subset of what's on the right. Make sense?