  IRAS 00500+6713 is a hypothesized remnant of a type Iax supernova SN 1181.
Multi-wavelength observations have revealed its complicated morphology; a dusty
infrared ring is sandwiched by the inner and outer X-ray nebulae. We analyze
the archival X-ray data taken by XMM-Newton and Chandra to constrain the
{angular radius}, mass, and metal abundance of the X-ray nebulae, and construct
a theoretical model describing the dynamical evolution of IRAS 00500+6713,
including the effects of the interaction between the SN ejecta and the intense
wind enriched with carbon burning ashes from the central white dwarf (WD)
J005311. We show that the inner X-ray nebula corresponds to the wind
termination shock while the outer X-ray nebula to the shocked interface between
the SN ejecta and the interstellar matter. The observed X-ray properties can be
explained by our model with an {ejecta kinetic} energy of $E_\mathrm{ej} =
(0.77 \mbox{--} 1.1)\times 10^{48}$~erg, an ejecta mass of $M_\mathrm{ej} =
0.18\mbox{--}0.53~M_\odot$, if the currently observed wind from WD J005311
started to blow $t_\mathrm{w} \gtrsim 810$ yr after the explosion, i.e.,
approximately after A.D. 1990. The inferred SN properties are compatible with
those of Type Iax SNe and the timing of the wind launch may correspond to the
Kelvin-Helmholtz contraction of the oxygen-neon core of WD J005311 that
triggered a surface carbon burning. Our analysis supports that IRAS 00500+6713
is the remnant of SN Iax 1181 produced by a double degenerate merger of
oxygen-neon and carbon-oxygen WDs, and WD J005311 is the surviving merger
product.
