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,