# reduction from SAT to approximate set cover

I read this neat result proved in the early 90s:

For any $$c>1$$, There's a poly time map from boolean formulas $$\varphi$$ to pairs $$K, \mathcal S$$ where $$K$$ is a positive integer and $$\mathcal S$$ is an instance of set-cover (i.e. $$\mathcal S = (U, \{T_1,...,T_n\}\subset 2^U)$$) such that

1. If $$\varphi$$ is satisfiable, $$\mathcal S$$ has an exact (i.e pair wise disjoint) cover of size $$K$$
2. If $$\varphi$$ is not satisfiable, $$\mathcal S$$ has no cover (exact or otherwise) of size smaller than $$c\cdot K$$.

The result is simple enough but its proof is based on the class MIP which i don't know much about and it's also (for someone of my limited background) strewn across a few different papers.

Has a more straightforward proof been discovered since 1993 or is there somewhere were i can read a more self contained proof of this or approximation results like this?