How is it possible that $L_1$ is $NP$?
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.