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We present a data-driven method for computing reachable sets for unknown nonlinear dynamical systems using a Koopman operator based approach. We find mixed-monotone decompositions for a class of Koopman lifted dynamics. The mixed-monotone system can be further embedded to a higher-dimensional dynamical model which is propagated in time deterministically. This allows us to find over-approximations of forward reachable sets that do not suffer from the curse of dimensionality. The proposed method can account for unknown nonlinear dynamics and allow a calculation of the conservative approximations of the reachable sets to a prefixed degree of accuracy in a computationally inexpensive manner. We demonstrate the efficacy of the proposed algorithm using an illustrative example with an unknown, nonlinear dynamical model, and compare it to CORA, a well-known existing method.