Hi guys!

I just worked on another simple set theory proof, and have come out with what seems like it might work alright, but while I was writing this I kind of felt like I was making some statements that perhaps needed to be more better justified. For example, in the second sentence of the proof, it kind of feels like I just pulled that out of thin air without justification.

If

, then

\

To show the left to right containment, let

. Then

. Note that there are two cases for

, either

or

. If

, then we have

and

, which implies that

\

. If

, then

, and so

\

. Therefore,

\

. To show the reverse containment, let

\

. Then

\

or

So we have two cases to consider. Suppose

\

. Then

and

. Since

,

. Suppose

. Then

. Note that by hypothesis

, and thus

clearly implies

. And so

. Since in either case

, we can conlude that

\

. Since we've shown both containments hold true, we can conlude that

\

.

Thanks for taking your time to read this,

James