Motivated by the recent multiplex framework of complex networks, in this work, we investigate if explosive synchronization can be induced in the multiplex network of two layers. Using nonidentical Kuramoto oscillators, we show that a sufficient frequency mismatch between two layers of a multiplex network can lead to explosive inter- and intralayer synchronization due to mutual frustration in the completion of the synchronization processes of the layers, generating a hybrid transition without imposing any specific structure-dynamics correlation.