Well, this is it.
The empty set is a subset of every set exactly because of the fact, that one doesn't need to verify, that every element of the empty set also belongs to a non-empty set.
If we have a property which no elements of a non-empty set have, we obtain the empty subset of this non-empty set:
[latex]\emptyset=\{x\in{M}|x\neq{x}\}[/latex]

...but as I eventually made clear for myself, it has nothing to do with a vector space...nothing can be defined on an empty set...from nothing comes nothing!