Prove that  is not a element of NOT-SELF I know this has been a question but based on a past experience, i thought i would rewrite it so i can get input and ask questions faster.

Suppose we have 
$$\text{NOT-SELF}=\{\langle M\rangle \mid  M \text{ is a Turing machine that does not accept }\langle M\rangle\}$$
the set of all machines that don't accept their own coding.

Let $Z$ be a TM such that $L(Z)$ is a subset of $\text{NOT-SELF}$. Prove that $\langle Z\rangle$ is an element of $\text{NOT-SELF}$.

My answer so far is very broad and I'm trying to improve it.

If $\langle Z\rangle$ is not an element of $\text{NOT-SELF}$, than $\langle Z\rangle$ can’t be an element in $L(Z)$. Since there’s a condition that $L(Z)$ is a subset of $\text{NOT-SELF}$,  this contradicts so $\langle Z\rangle$ must be an element of $\text{NOT-SELF}$.
Assume (i) $L(Z) \subseteq \text{NOT-SELF}$ and assume towards contradiction that (ii)
$\langle Z \rangle \notin \text{NOT-SELF}$.

Because of (ii) and (i) we get that $\langle Z \rangle \notin L(Z)$.

(if it was in $L(Z)$ and $L(Z)$ is a subset of $ \text{NOT-SELF}$, it must also be in $\text{NOT-SELF})$.

Thus, $Z$ does not accepts its own encoding. 
Thus, by the definition of $ \text{NOT-SELF}$ it should be that $\langle Z \rangle \in \text{NOT-SELF}$, which is a contradiction.

Also see the related question:
Can I construct a Turing machine that accepts only its own encoding?