A sealing system for a well valve includes a valve closure, a valve closure carrier, and seal elements. The valve closure includes a central bore. The valve closure carrier includes an annular, metal sealing surface in contact with the exterior of the valve closure to form a metal-to-metal seal with the valve closure. The valve closure carrier defines two flow holes, each of which communicates with the central bore when the valve closure is open to flow fluid, and is sealed from the central bore by the sealing surface when the valve closure is closed. The first seal element seals between the valve closure carrier and the valve closure, and surrounds the first flow hole. The second seal element seals between the valve closure carrier and the valve closure, and surrounds the first seal element and an end opening of the central bore when the valve closure is closed.