To confirm, you'd need to:
* Look under the vehicle: A separate transmission cooler will be a small, often finned, radiator-like unit somewhere in the front of the vehicle.
* Check the radiator: Many vehicles have the transmission cooler integrated within the radiator itself. This is less obvious to spot visually.
* Consult your owner's manual: This should specify if your specific trim level and engine configuration includes a transmission cooler.
Without access to a specific 2010 Silverado, I can't definitively answer.