Does $A\in B$ imply $A\subset B$? Question: Does $A\in B$ imply $A\subset B$ and does $A\in B$ and $B\in C$ imply $A\in C$?
I've been trying to find examples to get some intuition for this and I've come up with the following:
Example 1: Suppose that $A = \{1\}$, $B = \{\{1\},2\}$. I'd say that $A$ is an element of $B$ and $A$ is a subset of $B$.
Example 2: Suppose that $A = \{1\}$, $B = \{\{1\},2\}$ and $C = \{\{\{1\},2\},3\}$. Now $A\in B$, $B\in C$ but $A\not\in C$, right?
I think my confusion stems from the fact that I'm not sure how $B = \{\{1\},2\}$ vs $B = \{1,2\}$ determines whether $A$ is an element and/or a subset of $B$.

No. Taking $A=\{\emptyset\}$ and $B=\{A\} = \{\{\emptyset\}\}$, you have $A\in B$, but not $A\subseteq B$.

Furthermore, in Example $1$, $A=\{1\}$ is not a subset of the set $B=\{\{1\},2\}$, because it is not true that every element of $A$ is also an element of $B$. Specifically, $1$ is an element of $A$, but $1$ is not an element of $B$.
$B$ is a set with two elements, one of them is equal to $2$, the other one is equal to $\{1\}$, and $1$ is not equal to any of these two elements.
