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
  for A being epsilon-transitive set, B, C being Ordinal
  st A c= B & B in C holds A in C
proof
  let A be epsilon-transitive set, B, C be Ordinal;
  assume that
A1: A c= B and
A2: B in C;
  B c= C by A2,Def2;
  then
A3: A c= C by A1;
  A <> C by A1,A2,Th1;
  then A c< C by A3;
  hence thesis by Th7;
end;
