  In the class of admissible weak solutions, we prove a weak-strong uniqueness
result for the incompressible Euler equations assuming that the symmetric part
of the gradient belongs to $L^1_{\rm loc}([0,+\infty);L^{\rm
exp}(\mathbb{R}^d;\mathbb{R}^{d\times d}))$, where $L^{\rm exp}$ denotes the
Orlicz space of exponentially integrable functions. Moreover, under the same
assumptions on the limit solution to the Euler system, we obtain the
convergence of vanishing-viscosity Leray--Hopf weak solutions of the
Navier--Stokes equations.
