reserve p,p1,p2,q,r,F,G,G1,G2,H,H1,H2 for ZF-formula,
  x,x1,x2,y,y1,y2,z,z1,z2,s,t for Variable,
  a,X for set;
reserve M for non empty set,
  m,m9 for Element of M,
  v,v9 for Function of VAR,M;
reserve i,j for Element of NAT;

theorem Th81:
  not M |= x 'in' x & M |= 'not' x 'in' x
proof
  set v = the Function of VAR,M;
  not M,v |= x 'in' x by Th80;
  hence not M |= x 'in' x;
  let v;
  not M,v |= x 'in' x by Th80;
  hence thesis by ZF_MODEL:14;
end;
