Happy Mail lets you process every email in your inbox from newest to oldest - with the goal of reaching inbox zero.
Once you reach your goal, of no email in your inbox, you can sit back and relax - safe in the knowledge that all of your email has been handled.
Want to keep it in your inbox? Swipe down, it'll snooze for 10 minutes to let you make progress.
If you make a mistake then you can just tap the undo button and process the email again.
At the very least, you can use the handy "Archive All" button to save all of them, without having them in your inbox.