Statements that imply $\mathbf{P}\neq \mathbf{NP}$
A proof system for propositional logic is called polynomially bounded, if every tautology $\varphi$ has a proof in the system of length polynomial in the length of $\varphi$. 

The statement "There is no polynomially bounded propositional proof system" is equivalent to $\mathsf{NP} \neq \mathsf{co}\text-\mathsf{NP}$ by a classic result of Cook and Reckhow, so it implies $\mathsf{P} \neq \mathsf{NP}$.