Concerning contraction of the diaphragm:

When the diaphragm contracts, it becomes flatter The phrenic nerves that innervate the diaphragm come from high in the neck, that is, cervical segments 3, 4, and 5. Contraction of the diaphragm causes the lateral distance between the lower rib margins to increase and anterior abdominal wall to move out. The intrapleural pressure is reduced because the larger volume of the chest cage increases the recoil pressure of the lung.