Prove or disprove, If A ≤p B and B is NP-hard, then A is in NP-hard
Let A be the empty language. It is trivially reducible to, say 3SAT: transform any input into the formula $a \land \lnot a$. But only the empty language is reducible to the empty language, thus A is not NP-hard.

If P = NP, then any nontrivial language is NP-hard, thus it will be really hard to find counterexamples where both A and its complement are nonempty.