Hardness of problem related to number of subsets that satisfy a particular property
The problem is NP-hard: in fact, it is #P-complete.  Therefore, it is unlikely you'll be able to find an efficient solution for this problem.  The proof is by reduction from #Monotone-SAT.

As Ricky Demer explains, without loss of generality we can take the "predefined set" to be equal to the union of all properties.  (If the predefined set is larger than that, the answer to your problem is 0.  If the predefined set is smaller than that, then we can remove all elements that mention a property not found in the predefined set, and we get a new problem instance with the desired property.)  So let's focus only on this case.

Here is the reduction from #Monotone-SAT.  Suppose we have a CNF formula $\varphi$ over $n$ variables and with $m$ clauses, where all of the literals in each clause are positive (i.e., no variable is negated).  Then we can define an instance of your problem that corresponds to $\varphi$.  The problem instance has $n$ elements, one element for each variable, and $m$ properties, one property for each clause.  The set of properties of a variable is just the set of clauses that mention that variable.  Then the number of solutions to this problem instance is exactly the number of satisfying assignments to $\varphi$.

It follows that your problem is exactly as hard as #Monotone-SAT: any algorithm for your problem gives a solution to #Monotone-SAT.  However, it is known that #Monotone-SAT is #P-complete.  (In fact, we can make a stronger statement: #Monotone-2SAT is also #P-complete.)  It is widely believed that #P-complete problems cannot be solved in polynomial time.  Thus, there is unlikely to be any polynomial-time algorithm for #Monotone-SAT -- or for your problem.

That's the bad news.  Now one slight bit of hope.  There are off-the-shelf #SAT solvers out there that you could try applying to your problem.  See, e.g., https://cstheory.stackexchange.com/q/1295/5038 for some pointers.  Now you shouldn't expect them to be efficient -- they are subject to the same hardness result, and so they might be very slow for all but small problem instances.  However, if you absolutely must solve your problem in practice, this is something you could try.  There are also tools for approximate model counting, which could be used to compute an approximation/estimate for the number of solutions to your problem -- however, again, this is a hard problem and you shouldn't expect algorithms with a good worst-case running time.

My thanks to Ricky Demer for the key insight that enabled me to recognize the relationship to #Monotone-SAT.