Union of a language in P and a language in NP\P TRUE or FALSE Statement:


  Assume that $P \neq NP$. Then, for all languages $L_1$ and $L_2$, if $L_1$ is in P and $L_2$ is in NP but not in P then $L_1 \cup L_2$ is in NP but not in P. 


I have a doubt about the point: $L_1 \cup L_2$ is in NP but not in P.  

I know P is a subset of NP, then $L_1 \cup L_2$ is in NP (because $P \subseteq NP$), but the sentence "is in NP but not in P" 
it puzzles me.
From an Hint (by Yuval Filmus), the statement is FALSE because if $L_1=Σ^∗$ then $L_1 \cup L_2=Σ^∗$. I know $Σ^∗$ is in P (i.e., a TM which accepts all strings), and the statement is FALSE.