Consider the following premises. p_1 -> u. p_10. p_10 -> p_1. Can we infer ~u from them? If possible, name the inference rules being used at each step.
 From p_10 -> p_1, p_1 -> u we can infer p_10 -> u via transitivity. Finally, from p_10 -> u, p_10 we can infer u via modus ponens, which contradicts ~u. Therefore, the answer is no.