  A search for instability of nucleons bound in $^{136}$Xe nuclei is reported
with 223 kg$\cdot$yr exposure of $^{136}$Xe in the EXO-200 experiment. Lifetime
limits of 3.3$\times 10^{23}$ and 1.9$\times 10^{23}$ yrs are established for
nucleon decay to $^{133}$Sb and $^{133}$Te, respectively. These are the most
stringent to date, exceeding the prior decay limits by a factor of 9 and 7,
respectively.
