Say I have three sets $S_1, S_2, S_3$, and I want to show that they are equal. Is it enough to show that $S_1 \subset S_2$, $S_2 \subset S_3$, and $S_3 \subset S_1$?
I think yes:
(1) If $S_1 \subset S_2$ and $S_2 \subset S_3$, then $S_1 \subset S_3$. This result, together with $S_3 \subset S_1$, implies that $S_1 = S_3$.
(2) If, $S_1 = S_3$ and $S_1 \subset S_2$, then $S_3 \subset S_2$. This result, with $S_2 \subset S_3$ shows that $S_2 = S_3$.
(3) If $S_1 = S_3$ and $S_2 = S_3$, then $S_1 = S_2$.
So I think yes, but I'm planning to use this idea to prove something else, and I don't want to waste my time on a faulty premise! Thanks.