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We consider a coupled PDE-ODE system describing the motion of the rigid body in a container filled with the incompressible, viscous fluid. The fluid and the rigid body are coupled via Navier's slip boundary condition. We prove that the local in time strong solution is unique in the larger class of weak solutions on the interval of its existence. This is the first weak-strong uniqueness result in the area of fluid-structure interaction with a moving boundary.1 kinematic coupling condition in the literature is the standard no-slip boundary condition: the continuity of velocities of the fluid and the solids at the fluid-body interfaces. Such approach has been investigated by many authors, e.g. [3]-[8], [17,24] and references within. Despite lot of progress there are only few uniqueness results for weak solutions. The uniqueness of weak solution for the fluid-rigid body system in 2D was proven in [14]. The uniqueness of weak solution to 3D Navier-Stokes equation is a famous open problem (see e.g. [10]). However, it is well known that strong solution (i.e. solution with some extra regularity or integrability) is unique in a larger class of weak solution on interval of its existence [25]. These type of results are called weak-strong uniqueness results. The main result of this paper is to prove a weak-strong uniqueness result for the fluid-rigid body system. We also mention that in [6] the authors studied the motion of rigid body containing a cavity filled with the fluid and proved a weak-strong uniqueness result for that problem.However, it has been shown in [15], [16], [26] that the non-slip condition exhibits an unrealistic phenomenon: two smooth solids can not touch each other. The non-slip condition prescribes the adherence of fluid particles to the solid boundaries and, as a consequence of a regularity of the fluid velocity, permits the creation of fine boundary layer that does not allow the contact of the solids.Another method for coupling of the fluid and of the bodies admits the slippage of fluid particles at the boundaries, which is described by Navier's boundary condition. The first step in this direction of the study of Navier's condition was done by Neustupa, Penel [22], [23], who demonstrate that the collision with a wall can occur for a prescribed movement of a solid ball, when the slippage was allowed on both boundaries. We refer for a discussion of Navier's boundary condition to Introduction of [20]. In this last work a local in time existence result was demonstrated for the motion of the fluid and an elastic structure with prescribed Navier's condition on the boundaries. Motivated by these works, in this paper we study coupling via Navier's slip boundary condition. Also we mention the article [12] where a local existence up to collisions of a weak solution for a fluid-solid structure was proved. The existence of a strong solution in 2D case was proven by Wang [29].The global in time existence of the weak solution was proven in [2] for a mixed case, when Navier's condition was given on the solid boundar...

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