TITLE: Can sets $A,B,C$ satisfy $(A\in B \in C) \land (A \subseteq B \subseteq C)$?
QUESTION [19 upvotes]: We have a union of criteria: $$(A\in B \in C) \land (A \subseteq B \subseteq C)$$
Can such sets exist? 
I spent some time thinking about this and I came up with this example: 
$$A = \{ a\}$$ $$B = \{ \{ a\}, a  \}$$ $$C = \{ \{  \{a \}, a\}, \{a\}, a \}$$ 
Can this actually work?

REPLY [15 votes]: Perfect, you actually recovered the structure of the first nonzero ordinals: $1,2,3$. Starting with the set $a$ you applied the successor function $S(x):=x\cup\{x\}$ which is the first method for building bigger ordinals. Ordinals are $\in$-transitive. Id est $x\in y$ and $y\in z$ implies $x\in z$ which could seem strange at first glance, beating our level-idea of sets.