Is deciding if there is a set that is intersected with some given sets and that has at most one common element with other given sets NP-complete? Given 2 collections of ﬁnite sets $A_1,A_2,\ldots,A_m$ and $B_1,B_2,\ldots,B_n$, is there a set $T$ such that: $\left|T \cap A_j\right|\ge 1$, for $j = 1,2,\ldots,m$ and $\left|T \cap B_i\right|\le 1$, for $i= 1,2,\ldots,n$?

I tried doing this problem by considering it as a decision problem (pi).
Please suggest me if you have an approach.
You can reduce 3SAT to this problem by converting each variable $v_i$ to two elements $e_{i1}$ and $e_{i2}$ (one for its positive literal and one for its negative literal) and a set $B_i$ containing $e_{i1}$ and $e_{i2}$, and converting each clause $c_j=l_{j1}\vee l_{j2}\vee l_{j3}$ to a set $A_j$ containing elements representing $l_{j1},l_{j2},l_{j2}$. 

Now it's not hard to prove this problem is NP-complete using the reduction above.