The Collaborative International Dictionary of English v.0.48:

Inexistence \In`ex*ist"ence\, n. [Pref. in- in + existence.]
   [Obs.]
   (a) Inherence; subsistence. --Bp. Hall.
   (b) That which exists within; a constituent. --A. Tucker.
       [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Inexistence \In`ex*ist"ence\, n. [Pref. in- in + existence: cf.
   F. inexistence.]
   Lack of being or existence.
   [1913 Webster]