The invention refers to a teatcup part and a teatcup to be attached to a teat of an animal. The teatcup includes the teatcup part (1) and a connection part (2). The teatcup part is disengageably mountable to the connection part. The teatcup part (1) includes a shell (3) and a teatcup liner (4) having an upper opening (5) and a lower opening (6). The liner forms an inner space (11) for receiving the teat through the upper opening and a pulsation chamber (12) between the liner and the shell. The connection part includes a milk-discharging member for the discharge of milk from the teatcup, and a milk channel (35) extending between the lower opening of the inner space of the liner and the milk-discharging member. The connection part also includes a pulse-supplying member for the supply of a pulsating pressure to the pulsation chamber, and a pulse channel extending between the pulsation chamber and the pulse-supplying member.