reserve X for set;
reserve UN for Universe;

theorem Th81:
  for G being Grothendieck holds union G = G
  proof
    let G be Grothendieck;
    per cases;
    suppose G is empty;
      hence thesis by ZFMISC_1:2;
    end;
    suppose G is non empty;
      then reconsider G as non empty Grothendieck;
      now
        now
          let o be object;
          assume o in union G;
          then consider o9 be set such that
A1:       o in o9 in G by TARSKI:def 4;
          reconsider o9 as Element of G by A1;
          G is axiom_GU1;
          hence o in G by A1;
        end;
        hence union G c= G;
        now
          let o be object;
          assume o in G;
          then reconsider o9 = o as Element of G;
          o in {o9} in G by TARSKI:def 1;
          hence o in union G by TARSKI:def 4;
        end;
        hence G c= union G;
      end;
      hence thesis;
    end;
  end;
