How is it possible that $L_1$ is $NP$? The question is from my complexity-theory course.

The question

If $L_1,L_2,L_3 \in \lbrace 0,1 \rbrace^*$, and different from $\lbrace 0,1 \rbrace^*$ and from the empty language, prove that if $L_1 \leq_p L_2  \cap L_3$, where $L_2 \in NP$ and $L_3 \in P$ then $L_1 \in NP$

My answer

Due to the fact that $P \subseteq NP$, we know that $L_2 \in NP$ and that $L_1 \in P$. $L_2 \cap L_3$ is at most $L_3$ so $L_2 \cap L_3 \in P$ then I proved that $L_1 \in P$. But the question asks to prove that $L_1 \in NP$. Is it a joke? Am i missing something?
No, your proof is faulty.  You claim "we know that $L_1 \in P$", but that doesn't follow.  You haven't justified that claim, and I don't think it's true. It's not clear why you think that would be true, so it's hard to say why you went astray, but that's the first step that is wrong.  Also "is at most $L_3$ is not well-defined, and the intersection of a language in $P$ and a language in $NP$ is not necessarily in $P$.

I think you need to be more careful with your proofs: you have multiple steps where you make claims that simply don't follow from the previous steps.  I recommend you justify each step much more carefully, in more detail -- and if you can't, you should be suspicious of whether it is actually true.