If $\{x\}=\{z\}$ then $x=z$? I think that no. For example, Let $x=\{1,2,3\}$ and $z=\{1,2,3,4\}$. So, $\{\{1,2,3\}\}=\{\{1,2,3,4\}\}$. Yet, $x\neq z$.

Can you explain?
By definition two sets are equal if and only if they have the same members: $x=y$ if and only if $\forall z(z\in x\leftrightarrow z\in y)$. In your example the sets $\{x\}$ and $\{z\}$ are assumed to be equal, so they must have exactly the same elements. The only element of $\{x\}$ is $x$, and the only element of $\{z\}$ is $z$, so it must be the case that $x=z$.

As was noted in the comments, your example is incorrect: if $x=\{1,2,3\}$ and $z=\{1,2,3,4\}$, then $x\ne z$, so $\{x\}\ne\{z\}$.
$x$ consists of $4$ elements, $z$ consists of $3$ elements. Obviously, they cannot be equal.