NP $\subsetneq$ EXP?
Strictly contained means $\subsetneq$, i.e. it is the definition, it is not a result. So what you are saying is 
$$\mathsf{NP} \subsetneq \mathsf{ExpTime} \implies \mathsf{NP} \subsetneq \mathsf{ExpTime}$$
 which is trivially true.

If you are asking if $\mathsf{NP}\subsetneq \mathsf{ExpTime} $ then the answer is: it is unknown. 

You may want to check the Wikipedia article about Exponential Time Hypothesis which many experts believe is true.