# prove that if A is a subset of B, B is a subset of C, and C is a subset of A, then A=B and B=C

To prove A=B, I must prove that A is a subset of B and B is a subset of A. A is a subset of B is already given. So all that is left is to prove B is a subset of A.

Is it suffice to say that since A is a subset of B, B is a subset of C, and C is a subset of A, by transitive property B is a subset of A.

• Yes. It is. Adding more characters so I can post comment.
– user142299
Apr 17, 2014 at 1:55
• Do you mean I should add more characters on my post so you can post comments? Apr 17, 2014 at 1:56
• No I mean that I had to add characters to post my comment. :) Try posting a comment with one word it won't work.
– user142299
Apr 17, 2014 at 1:57
• Probably you need to prove the transitive property. Or have you already seen a proof of it? Apr 17, 2014 at 1:57
• No I have not. That's why i asked because I feel that simply stating its transitive property is not really a proof Apr 17, 2014 at 1:59

Check transitivity: $$A\subseteq B\subseteq C\implies A\subseteq C$$
Check antisymmetry: $$A\subseteq C\subseteq A\implies A=C$$
If you are given that $A\subseteq B\subseteq C\subseteq A$, then yes you can. To be more specific, given any $x\in B$, $x$ is also in $C$, and also in $A$, so $B\subseteq A$. Use a simlar argument to show that $B=C$.