The classical three-stage model of membrane solubilization, including mixed membranes, membrane-micelle coexistence, and mixed micelles, is not applicable to demixed, domain-forming membranes and must, therefore, fail to describe the phenomenon of detergent-resistant membranes (DRMs). In lack of a quantitative model, it has often been assumed that ordered, detergent-depleted domains are inert, whereas fluid domains are solubilized. We establish a quantitative model based on equilibrium thermodynamics that is analogous to the three-stage model but comprises three components (two lipids and one detergent) in four phases (liquid-ordered and liquid-disordered membranes, micelles, and detergent in aqueous solution). For a given set of total concentrations and input parameters (initial abundance of ordered domains, solubilization boundaries of the pure lipids, etc.), it serves to calculate the phase boundaries and partial concentrations of all components in all phases. The results imply that the abundance and composition of ordered domains may vary substantially upon addition of detergent, both before and during solubilization of the fluid phase. It seems that gel-phase or order-preferring lipids are thermodynamically "resistant" regardless of the presence of a second, fluid phase. However, thermodynamic or kinetic resistance is not sufficient for obtaining DRMs because the resistant particles may be too small to be isolated. Cholesterol may be crucial for rendering the fragments large enough and, furthermore, enhance the formation of ordered domains by nonideal interactions with the detergent.