reserve N,M,K for ExtNat;
reserve X for ext-natural-membered set;

theorem
  for A being Ordinal holds A <> 0 & A <> 1 iff A is non trivial
proof
  let A be Ordinal;
  hereby
    assume A1: A <> 0 & A <> 1;
    then A in 1 or 1 in A by ORDINAL1:14;
    then A2: 1 in A by A1, CARD_1:49, TARSKI:def 1;
    0 in 1 by CARD_1:49, TARSKI:def 1;
    then 0 in A by A2, ORDINAL1:10;
    hence A is non trivial by A2, ZFMISC_1:def 10;
  end;
  assume A is non trivial;
  then consider x, y being object such that
    A3: x in A & y in A & x <> y by ZFMISC_1:def 10;
  A4: card {x,y} c= card A by A3, ZFMISC_1:32, CARD_1:11;
  card A c= A by CARD_1:8;
  then card {x,y} c= A by A4, XBOOLE_1:1;
  then {0,1} c= A by A3, CARD_2:57, CARD_1:50;
  then 0 in A & 1 in A by ZFMISC_1:32;
  hence thesis;
end;
