We do not work for righteousness any more than we worked for unrighteousness. This is the reasoning behind grace. Paul proves the Gospel is indeed valid and effective. This helped believers realize they did not work for salvation. Moreover the law entered, that the offence might abound.
Grace abounds more than sin.
See verses 15-19 again, how that Jesus' act abounded greater than Adam's act. The law came to make the effects of Adam's transgression more apparent to mankind. It made people truly desire to be free of that sin. It showed how horrible sin was. And the commandment, which was ordained to life, I found to be unto death.