In this work, we show how controlled robustly forward invariant sets for systems with disturbances are efficiently identified via the application of the mixed monotonicity property. A mixed monotone system can be embedded in a related deterministic embedding system with twice as many states but for which the dynamics are monotone; one can then apply the powerful theory of monotone dynamical systems to the embedding system to conclude useful properties of the initial mixed monotone system. Using this technique, we present a method for verifying state-feedback controllers against safety (set invariance) constraints, and our approach involves evaluating a control barrier function type condition that requires the vector field of the embedding system to point into a certain southeast cone. This approach also facilitates the construction of runtime assurance mechanisms for controlled systems with disturbances, and we study system safety in the presence of state uncertainty as well. The results and findings of this work are demonstrated through two numerical examples where we study (i) the verification of a controlled spacecraft system against a safety constraint, and (ii) the formation of a runtime assurance mechanism that functions in the presence of uncertain state measurements.