Large magnetostriction and first-order phase transition induced by ultrahigh magnetic fields in V$_6$O$_{13}$

V$_6$O$_{13}$ exhibits the metal-insulator transition (MIT) at around 150 K with the formation of vanadium-vanadium (V-V) dimers. We measured the magnetostriction of V$_6$O$_{13}$ along the crystallographic $b$-axis in ultrahigh magnetic fields up to 186 T at various temperatures. An abrupt and large negative magnetostriction of $\Delta L / L \sim 10^{-3}$ was observed above 120 T below 150 K which is attributed to the collapse of the V-V dimer. A pronounced hysteresis exceeding 50 T was observed, indicating a first-order phase transition. The field-induced collapse of the V-V dimer is expected to be accompanied by the insulator-to-metal transition. The robustness of the insulating phase against magnetic fields was discussed based on thermodynamics and the comparison with the case of VO$_2$.