The resonant trident pair production process in the collision of ultrarelativistic electrons with a strong electromagnetic wave was theoretically studied. Under resonant conditions, the intermediate virtual gamma-quantum became real. As a result, the original resonant trident pair production process effectively split into two first-order processes by the fine structure constant: the electromagnetic field-stimulated Compton effect and the electromagnetic field-stimulated Breit–Wheeler process. The kinematics of the resonant trident pair production process were studied in detail. It was shown that there are two different cases for the energies and outgoing angles of the final particles (an electron and an electron–positron pair) in which their quantum entanglement is realized. In the first case, energies and outgoing angles of the final ultrarelativistic particles are uniquely determined by the parameters of the electromagnetic field-stimulated Compton effect (the outgoing angle of the final electron and the quantum parameter of the Compton effect). In the second case, energies and outgoing angles of the final particles are uniquely determined by the electromagnetic field-stimulated Breit–Wheeler process (the electron–positron pair outgoing angle and the Breit–Wheeler quantum parameter). It was shown that in a sufficiently wide range of frequencies and intensities of a strong electromagnetic wave, and in the case of ultrarelativistic initial electrons, the differential probability of the resonant trident pair production process with simultaneous registration of the outgoing angles of the final particles can significantly (by several orders of magnitude) exceed the total probability of the electromagnetic field-stimulated Compton effect.