theorem Th21:
  aleph A = aleph B implies A = B
proof
  assume
A1: aleph A = aleph B;
A2: now
    assume B in A;
    then aleph B in aleph A by Th20;
    hence contradiction by A1;
  end;
  now
    assume A in B;
    then aleph A in aleph B by Th20;
    hence contradiction by A1;
  end;
  hence thesis by A2,ORDINAL1:14;
end;
