ASW > Zspread for a premium bond?

Apologies if this is the wrong subforum. I am reading in one of my notes:

Premium bonds

  • Par ASW > Zspread

Discount bonds

  • Zspread > Par ASW

I would have logically thought that the opposite is true. Under a Par ASW, I was thinking that since the investor pays par and receives the dirty price upfront, if the bond is trading at a premium, this upfront has to be subtracted form the ASW over the life of the swap, thereby reducing it. This reduction then logically leads me to believe the par ASW would be lower than the Zspread. For a discount bond, the logic is the same: I would have thought that the upfront cost to the investor is added over the life of the swap to the ASW, making it greater than the Zspread.

Could someone please shed some light on this? Thank you.