Conservation of Momentum in a Single Direction

The Law of Conservation of Momentum states that the total linear momentum does not change in any closed system not subject to external forces. However, is it true that the momentum in, say, the x-direction does not change in any closed system not subject to forces in the y-direction?

I feel that it is indeed true since y-directional forces do not change the x-directional velocity, thus conserving momentum in the x-direction.