There is an interpretation of your inequality using simplicial complex and $h$-vector. Namely, let $\Delta$ be the set of all $P-S$ with $S$ in your set. Then $\Delta$ is a collection closed under inclusion, hence a [simplicial complex](https://en.wikipedia.org/wiki/Abstract_simplicial_complex). 

The invariant you are interested in is, up to sign, the alternating sum of $f_i$: the number of $i$-dim faces of $\Delta$. Then, by a well-known formula, it is equal to $|h_d|$, where $h_i$ forms the [$h$-vector](https://en.wikipedia.org/wiki/H-vector) of $\Delta$, and $d-1$ is the dimension of $\Delta$. 

The right hand side of the inequality you are interested in is the number of facets (maximal elements) of $\Delta$. As people have pointed out, in general the inequality you want does not hold. 

However, I would like to point out that it is likely to hold certain extra topological/homological  assumptions. For instance, if $\Delta$ is Cohen-Macaulay, then all the $h_i$ are non-negative, and the number of facets is $f_{d-1}=\sum_{i\geq 0} h_i\geq h_d$, which is what you need.

One can prove certain inequality for $h_i$ under weaker conditions. For instance, I think your inequality still holds under Serre's condition $(S_{d-1})$ (a notch weaker than Cohen-Macaulay).  I discussed some of them in a [recent talk](https://www.msri.org/seminars/25063) (but it is perhaps a bit algebraic).