List of theorems stating that P does not equal NP if and only if I think it would be a good idea to make a list of theorems stating that P does not equal NP if and only if such and such exits, some complexity class is contained in another complexity class and so on and so forth.
Here is a one:

Mahaney's Theorem: There is no sparse NP-complete set if and only if $P \ne NP $
(under Karp reduction).

Another one is:

$P \ne NP$ if and only if $P \ne PH$