  Fragmentation cross section of $^{28}$Si + $^{9}$Be reaction at 75.8 MeV/u
was analyzed for studying the decay mode of single-proton emission in $^{21}$Al
(the proton-rich nucleus with neutron closed-shell of $N = 8$ and $T_z =
-5/2$). With the comparison between the measured fragmentation cross section
and the theoretical cross section produced by EPAX3.1a for the observed nuclei
(i.e. $^{20}$Mg, $^{21}$Al and $^{22}$Si), the expected yield for a particle
stable $^{21}$Al was estimated. With the exponential decay law, an upper limit
of half-life of $13$ ns was determined. Using the single-proton penetration
model, the upper limit of single-proton separation energy of $-105$ keV was
deduced. This deduced mass limit agrees with the microscopic calculation based
on nucleon-nucleon (NN) + three-nucleon (3N) forces in $sdf_{7/2}p_{3/2}$
valence space, which indicates the importance of 3N forces in $^{21}$Al.
