Let me contextualize the title to make myself clearer: thanks to Cook–Levin theorem, there is a well known, and easy to understand way to prove that $P = NP$. It is known that if one can prove that an algorithm solves an NP-complete problem, and has polynomial complexity, that implies that $P = NP$, and the problem is solved.
It is almost like a guide for researches on the field on what to focus the effort: they "just" have to find a polynomial algorithm for SAT or Traveling Salesman or some other NP-complete or NP-hard problem, and $P$ vs. $NP$ is solved.
Are there any results of the kind for $P \neq NP$? Some thing or property that if found or proven will imply in $P \neq NP$? Reading this blog post or the Wikipedia page on P versus NP makes me believe that there isn't, and that a proof scheme for $P \neq NP$ is much more elusive and/or harder to understand than for $P = NP$.
Note that $P$ and $NP$ are defined in terms of Turing Machines. $P$ is the set of languages that can be decided by a deterministic Turing Machine in polynomial time. The class $NP$ is the set of languages where string membership can be verified in polynomial time by a non-deterministic Turing Machine.
So it would suffice to show that no deterministic Turing Machine can decide an $NP$-Complete language in polynomial time, if you want to show that $P \neq NP$. Impossibility results are harder to show than existence results, but there are cases of impossibility results being shown. The halting problem is a particularly relevant example. No Turing Machine exists to decide $L_{HALT} = \{ \langle M_{i}, w_{i} \rangle : M_{i}$ halts on $w_{i} \}$.