Should I get rid of this tree?

[Jossi S.]

Hope this is the right forum - I have a question for the gardening and landscaping experts in the BP community. 

This tree/shrub is planted very close to a field stone foundation. I cut it way back in the Fall, but apparently that only made it stronger. After looking at lots of leaf pictures I think it might be a Siberian Elm, but not totally sure. Location is in MA.  

Should I remove it? If so, can I just cut it down to the soil, or would it just grow back again?