Minimal functionally complete operator sets with XOR
XOR must be included (this is given).
XOR and AND can be shown to be complete. (OP has done this)
XOR and AND is only two elements (by inspection).
To be more minimal, you would have to remove AND.
The only way to remove AND and maintain completeness is to implement AND using only XOR.

Thus, stating AND cannot be implemented using only XOR is exactly equivalent to saying that {AND, XOR} is the minimal set containing XOR. It may not be the ONLY minimal set, but it is one minimal set.

To generate a rigorous proof, see the work of Emil Post. He apparently discovered the properties necessary for completeness. So if you can use his theorem, you can show that XOR is not complete by itself using his work.