You are looking for "Ripple Trim Next Edit To Playhead" and "Ripple Trim Previous Edit To Playhead". Set them manually in the Keyboard Shortcuts dialog.
Be ware that there is a bug that will remove any transitions at the cut point in CS6 when using those commands. So it may be useful or just annoying. This is fixed in Premiere Pro CC.
Or, use Ctrl+K + select the clip you want to delete and press Shift+Delete to ripple it away in one go.
Or, set an In Point and an Out Point and use Extract to ripple away the media between then In/Out Point.