I'm looking through the lecture notes for my course on modal logic and am having a hard time understanding why it is that U, the class of all Universal frames, is a subset of E, the class of all Equivalence frames. I understand that all universal relations are equivalence relations and that not all equivalence relations are universal, but then do not see how the following assertion follows:
"If φ is valid in E, then since U is a subset of E, φ is valid in U." Wouldn't it be possible that φ is valid in U but not in E, since not there are some relations in E that are not in U?
I guess I'm thinking of U being a subset of E as the implication, "If U then E," which perhaps is not the correct way of thinking about it? Alternatively, I suppose we can consider the set of binary relations in each frame; then, if for x to be a subset of y is for all of the members of x to be members of y, wouldn't we have that E is a subset of U? As all of the binary relations in E are in U but U has many more relations?
Thanks in advance for any insight into the question itself and into the definitions involved.