reserve E,X,Y,x for set;
reserve A,B,C for Subset of E;
reserve x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 for Element of X;
reserve x for object;

theorem
  for X being non trivial set, p being set ex q being Element of X st q <> p
proof
  let X be non trivial set, p be set;
  X \ { p } is non empty by ZFMISC_1:139;
  then consider q being object such that
A1: q in X \ { p };
  reconsider q as set by TARSKI:1;
  X \ { p } c= X by XBOOLE_1:36;
  then q in X by A1;
  then reconsider q as Element of X by Def1;
  take q;
  thus thesis by A1,ZFMISC_1:56;
end;
