Example 3

Are you sure you mean "A belongs to B"? If yes, it would mean set A is an element of set B. That is, $B=\left\{\left\{A\right\}, other elements of set B\right\}$. Then, since $B\subset C,$it means that set A is also an element of set C. That is, $C=\left\{\left\{A\right\}, other elements of set C\right\}$. So, set A cannot be a subset of set C, as it is an element of set C.

For example, say $$\begin{gathered} A=\left\{1, 2, 3\right\} \\ B=\left\{\left\{1, 2, 3\right\}, 4, 5\right\} \\ C=\left\{\left\{1, 2, 3\right\}, 4, 5, 6, 7\right\} \end{gathered}$$

Here, $A\in B \mathrm{and} B\subset C. \mathrm{So}, A\in C \mathrm{and} A \mathrm{is} \mathbf{not}\subset C$

However, if you mean "A is a subset of B and B is a subset of C", then yes, it is true that A is a subset of C. That is, $if A\subset B and B\subset C, then A\subset C$.