SSUM is the same as the Subset Sum Problem with the only additional requirement is all the numbers must be unique in the subset.
To prove it's NP complete, the verifier is quite easy to construct being the same as one for the Subset Sum except you add the additional requirement of making sure all numbers are unique.
With the reduction, I assume you can use the same reduction as before from 3SAT, I'm just figuring out a way when evaluating that reduction, to determine if two or more numbers are the same. I'm using the 3SAT reduction in Sipser with the table of columns and rows. If one needs any more information I would be happy to provide.