reserve X,Y,Z,X1,X2,X3,X4,X5,X6 for set, x,y for object;
reserve a,b,c for object, X,Y,Z,x,y,z for set;
reserve A,B,C,D for Ordinal;

theorem Th10:
  A in B or A = B or B in A
proof
  assume that
A1: not A in B and
A2: A <> B;
  not A c< B by A1,Th7;
  then not A c= B by A2;
  then consider a be object such that
A3: a in A & not a in B;
  a in A \ B by A3,XBOOLE_0:def 5;
  then consider X such that
A4: X in A \ B and
A5: not ex a being object st a in A \ B & a in X by TARSKI:3;
A6: X c= A by A4,Def2;
  now
    let b be object;
    assume
A7: b in X;
    then not b in A \ B by A5;
    hence b in B by A6,A7,XBOOLE_0:def 5;
  end;
  then X c= B;
  then
A8: X c< B or X = B;
  ( not X in B)& X is Ordinal by A4,Th9,XBOOLE_0:def 5;
  hence thesis by A4,A8,Th7;
end;
