Does this principle have a name? $A\subset B\subset A \implies A=B$

Question

I have a proof about some particular sets, which starts with something like $A \subset B$, and after a sequence of steps I show finally that $B \subset A$.

The proof from a high level is something like $A \subset B = C = D = E = F \subset A$, therefore $A=B$

But we know that if $A\subset B$ and $B \subset A$ then $A=B$.

QUESTION: Does this principle have a name?

I don't think it's the definition of set equality per se, but rather something which is equivalent to it. Neither do I think that this is the axiom of extensionality, yet again it is closely related to it.

Answer 1

The inclusion $\subset$ - relation is "anti-symmetric":

$$ A \subset B, B \subset A \implies A=B.$$

Answer 2

Do you mean $\subseteq$ instead of $\subset$, proper subset?
Such theorems are called squeeze theorems.
For example, a <= b and b <= a implies a = b.