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 $B$.

Your question gives an occasion to emphasize the distinction between : 
(1) not implying that proposition P is true 
and 
(2) implying that proposition P is false ( not-true). 
The fact that A is a member of B certainly does not imply that A is included in B. 
Thousands of examples could be found. 
But, precisely, due to the fact that this non-implication is so often  confirmed, we tend to think that : "A belongs to B" implies that "A is not a subset of B". 
And surprisingly, when one discovers the construction of natural numbers, one has to recognize that this induction is false. 
For  example :  2 is defined as {0,1} and 3 is defined as {0,1,2}. The number 2 is a member of the number 3. But is as also a subset of 3.  
