TITLE: Is an element of a set that is contained within a larger set also an element of the larger set?
QUESTION [0 upvotes]: Suppose you have A = {1, 2, 3} and B = {4, 5, 6}.
Now suppose you have set C = {{A, B}}.
Of course, 3 ∈ A.
My question is: is it also the case that 3 ∈ C? C seems to contain 3 in some sense, but is it correct to say that 3 is element of C?
Edit: abiessu points out that I have written that A is an element of a set which is an element in C. This is not what I intedned to write, it was an error on my part. Let me now correct that. I meant to say that C = {A,B}.

REPLY [0 votes]: To achieve one of the goals stated in the comments, it is possible to adjust the language slightly so that the repetition in "$3$ is an element of an element of $C$" is not necessary.  For example, the word "contains" is ambiguous enough within set theory that we could provide our own meaning, perhaps as follows:
Let "$U$ is a descendant of $V$" mean that there is a finite sequence $E_0,E_1,E_2,\dots,E_{n-1},E_n$ such that $U\in E_n\in E_{n-1}\in \dots\in E_2\in E_1\in E_0\in V$.  Further, when either $K\in V$ or $K$ is a descendant of $V$ we say that $V$ contains $K$.
In the context of the sets in the question, we could then say that $C$ contains $3$.  Note that this definition of "contains" is different from "has subset".  It should be straightforward to prove that this definition is transitive.
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