reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;

theorem
  a <> b implies ord-type {a,b} = 2
  proof
    assume a <> b; then
A1: card {a,b} = 2 by CARD_2:57;
    a c= a\/b & b c= a\/b by XBOOLE_1:7; then
    a in succ(a\/b) & b in succ(a\/b) by ORDINAL1:22; then
A2: {a,b} c= succ(a\/b) by ZFMISC_1:32; then
    On {a,b} = {a,b} by ORDINAL3:6;
    hence thesis by A1,A2,CARD_5:36;
  end;
