Problem 11.6. Prove Proposition 11.16 When I = {A1, A2,.. ., Ax} is a finite set we may use the sim- plified notation A1, A2, . . ., Ak + B for F + B, in particular A + B means that {A} - B. Note that if I' r A and Ar B, then I' r B. It follows also that if A1, . . ., An + B and I + A; for each i, then Fr B. Proposition 11.16. The following are equivalent. 1. I is inconsistent. 2. Ir A for every sentence A. 3. IF A and I - -A for some sentence A. Proof. Exercise