I don't understand the definition of rank of a set. The definition I have is:
For any set $V_{\infty}$, the rank of $x$, written $\rho(x)$, is the least ordinal $\alpha$ for which $x \in V_{\alpha^+}$
$V_{\infty}$ is the class of all sets which are in at least one of the $V_{\alpha}$s. I've found here https://ncatlab.org/nlab/show/cumulative+hierarchy a few examples, and in particular that the rank of $\{1\}$ is $2$, but how? $\rho(\{1\})$=$\rho(\{\{\emptyset\}\})$, and its not true that $2=\{\emptyset,\{\emptyset\}\}$ is the least ordinal for which $\{\{\emptyset\}\} \in V_{2^+}=V_3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$ what am i missing?
$\{\{∅\}\}∉V_0=∅$
$\{\{∅\}\}∉V_1=\{∅\}$
$\{\{∅\}\}∉V_2=\{∅,\{∅\}\}$
$\{\{∅\}\}∈V_3=\{∅,\{∅\},\{\{∅\}\},\{∅,\{∅\}\}\}$
So, the first stage where $\{1\}$ appear is $3=2+1$, so the rank of $\{1\}$ is $2$