Join two meshes at the intersection within edit mode Problem

Curiosity question here, I have the problem where I have created two meshes have inside the same object (so we are inside edit mode) with the purpose of combining both of them as one joined mesh

For example, as illustrated here:



Question

In the picture, I have two mesh object that are disjoint, One 'large shell' mesh and rectangular square mesh. I wish to join these two faces as I showed in the picture and then clean up the unneeded vertices. Is there a way to do this within edit mode?

Cheers
Press "SPACE", select search, type "intersect".
Click "Intersect (Boolean)". 
Select "Union" from the dropdown menu.
Done.