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.
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