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?
 A: 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 $\{x\}\ne\{z\}$.
A: 
Notice that in order to prove that two set $A$ and $B$ are equal, we need to show: $$A\subseteq B~~~~ \text{and}~~~~ B\subseteq A$$

In this case we have the set $x=\{1,2,3\}$ and $z=\{1,2,3,4\}$.
We can clearly see that $x \subset z$, since every element of $x$ is contained in $z$. In order to show equality, we now also need $z\subset x$.
However, since $4 \in z$, but $4 \not\in x$ we have that $z \not\subset x$. 
So $x \neq z$.
A: I am going to hazard a guess that when the OP says

Let x={1,2,3} and z={1,2,3,4}. So, {{1,2,3}}={{1,2,3,4}}

that he or she is confusing "equal sets" with "equal size sets".  It is definitely true that $\{ \{1,2,3 \}\} $ and $\{ \{1,2,3,4\}\}$ are both sets containing a single element, and they are therefore equal in size.  But they are not equal sets.  "Equal sets" means "the same sets", which means "sets containing the same elements".  If you want to indicate that the two sets are the same size, use the notation $ \left| \{ \{1,2,3 \} \} \right| = \left| \{ \{ 1,2,3,4 \} \} \right|$.
A: That is correct. x $\neq z$.
$$x=\{1,2,3\}\  and\ z=\{1,2,3,4\}\ then\ x \neq z.
$$
But because all of $x$ is in $z$ and $z$ has items $x$ does not have, $x$ $\subseteq$ $z$. Meaning x is a proper subset of $z$.
A: By extensionality of sets, there is an equivalence
$$\{ x \} = \{ z \} \quad \Longleftrightarrow \quad \forall t: (t \in \{ x \} \Leftrightarrow t \in \{ z \}) $$
By definition of set enumeration notation, there is an equivalence
$$ a \in \{ b \} \quad \Longleftrightarrow \quad a = b $$
Substituting this into the above equivalence,
$$\{ x \} = \{ z \} \quad \Longleftrightarrow \quad \forall t: (t = x  \Leftrightarrow t =z) $$
The right hand side is equivalent to $x=z$. But if you need to see that proved, I will instead prove something simpler: by substituting $t = x$, we get
$$ \forall t: (t = x  \Leftrightarrow t =z) \implies (x=x \Leftrightarrow x=z) \implies x=z $$
and consequently,
$$\{ x \} = \{ z \} \quad \implies \quad x=z $$
The other direction
$$ \{ x \} = \{ z \} \quad \Longleftarrow \quad x=z $$
is proven by substitution, thus
$$ \{ x \} = \{ z \} \quad \Longleftrightarrow \quad x=z $$

Your error is that $\{\{1,2,3\}\} \neq \{\{1,2,3,4\}\}$.
A: $x$ consists of $4$ elements, $z$ consists of $3$ elements. Obviously, they cannot be equal.
