Yes, this build should work.
However, it would be nice if you could link to all the parts, as it is a bother to google them.
What exactly is this computer for?
The processor (3570K) is really powerful, and you only need an aftermarket cooler if you intend to do serious overclocking. The GPU (HD 6870) is just an average card, and will hold you back if you intend to play new games at high settings.
Will the SSD be your only data storage device?