Prove or Give counter example:

kind of a long proof. But I was wondering if anyone could help check this for me.

I proved that

and

. So

was a subset of

.

When trying to prove that

was a subset of

I reached a contradiction, well what i believe was a contradiction. I found that

and

. Im just not sure if this is a contradiction or not. If its a contradiction, well im done, if it is not a contradiction I guess I would just continue on to prove it. Im just not sure if the last part of the proof is a contradiction.

Any help with this proof is greatly appreciated, and im sorry if this is somewhat unclear.