reserve X,Y,Z,X1,X2,X3,X4,X5,X6 for set, x,y for object;
reserve a,b,c for object, X,Y,Z,x,y,z for set;
reserve A,B,C,D for Ordinal;

theorem Th18:
  A in succ C iff A c= C
proof
  thus A in succ C implies A c= C
  proof
    assume A in succ C;
    then A in C or A in { C } by XBOOLE_0:def 3;
    hence thesis by Def2,TARSKI:def 1;
  end;
  assume
A1: A c= C;
  assume not A in succ C;
  then A = succ C or succ C in A by Th10;
  then
A2: succ C c= C by A1,Def2;
  C in succ C by Th2;
  then C c= succ C by Def2;
  then succ C = C by A2;
  hence contradiction by Th5;
end;
