I am reading a Set Theory book by Kunen. He presents first-order logic and claims that if a set of sentences in inconsistent, then it proves every possible sentence. Since he does not explicitly specify the inference rules, I became curious as to how fundamental this property of inconsistent systems is.
So my question is what is the simplest proof, with the least use of assumptions, of the vague claim that "inconsistent systems can prove anything" - in particular I'm interested in the assumptions about the system needed to prove this - is it true only for first order logic? Only for first order logic with the "standard" rules of inference (Modus ponens and GEN)? Or is it such a basic truth that it can be proved for every "reasonable" proof system (and what is "reasonable")?