The valve that you have looks to be if the web photo is correct a 5600, and there is a little white shoe that depending on the lbs setting will hold the brine piston open for a timed refill.
I use the float assembly in the brine tank as the second back up to stop water flow to the brine tank..
The float in the brine tank is often the back up to stop water flow, the air check at the bottom stops air from getting into the system when the water level in the brine tank gets to the low point.