# Is the set $\{1\}$ a member, and not a subset of the power set of $\{1, 2\}$?

Is the set $\{1\}$ a member, and not a subset of the power set of $\{1, 2\}$?

I just want to be sure I am not making a mistake here, my reasoning:

A is a subset of B if for every $x \in A$, $x \in B$

However, the element in $\{1\}$ is simply $1$. But the power set of $\{1, 2\}$ is the set $\{\{\},\{1\},\{2\},\{1,2\}\}$ and this power set has no such element, it does however have an element that is the set containing the element $1$. So $\{1\}$ is an element of this power set, but NOT a subset, since the only element in $\{1\}$ does not appear in this power set.

Is my reasoning correct?

-
$\{1\}$ is a member of the power set; $\{\{1\}\}$ is a subset of the power set.