Let $X=\{A,\{c,d\}^2\}$ be a set. Then, \begin{gather} \{c,d\}^2=\{c,d\}\times\{c,d\}=\{(c,c),(c,d),(d,c),(d,d)\} \end{gather}
My question is: does the pair $(c,d)$ belong to the set $X$? In other words, does an element of a Cartesian product belong to a set containing that product?
The motivation here is to better understand the property of what it means to be an element of a set and to learn more about the connection between a cartesian product and the elements of the set generated by the cartesian product of two valid sets.
In your set $X=\{A,\{c,d\}^2\}$, there are two elements. $A$ and $\{c,d\}^2$. Therefore, your question of whether or not the pair $(c,d)$ is an element of $X$ depends on how we define the set $A$.
If $A = (c,d)$, then because $A$ is an element of $X$, we know that $(c,d)$ is an element of $X$. However, if $A$ is defined to be anything else, then the answer is no, $(c,d)$ is not an element of $X$.
More generally, $A$ is an element of $\{A\}$, but is not an element of $\{\{A\}\}$. The only element of $\{\{A\}\}$ is $\{A\}$ (which is not equal to $A$).