reserve X for set;
reserve UN for Universe;

theorem Th31:
  for u,v being Element of UN holds
  { [u, {} ], [v, {{}} ] } = [: {u}, { {} } :] \/ [: {v}, { {{}} }:]
  proof
    let u,v be Element of UN;
    set S1 = { [u, {} ] , [v, {{}} ] },
        S2 = [: {u}, { {} } :] \/ [: {v}, { {{}} }:];
    now
      now
        let o be object;
        assume o in S1;
        then o = [u,{}] or o = [v,{{}}] by TARSKI:def 2;
        then o in [:{u},{{}}:] or o in [:{v},{ {{}} }:] by ZFMISC_1:28;
        hence o in S2 by XBOOLE_0:def 3;
      end;
      hence S1 c= S2;
      now
        let o be object;
        assume o in S2;
        then o in [: {u}, { {} } :] or o in [: {v}, { {{}} }:]
          by XBOOLE_0:def 3;
        then o in {[u,{}]} or o in {[v,{{}}]} by ZFMISC_1:29;
        then o = [u,{}] or o = [v,{{}}] by TARSKI:def 1;
        hence o in S1 by TARSKI:def 2;
      end;
      hence S2 c= S1;
    end;
    hence thesis;
  end;
