reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem Th47:
  X is epsilon-transitive implies ex A st Tarski-Class X c= Rank A
proof
  assume
A1: X is epsilon-transitive;
  assume
A2: not Tarski-Class X c= Rank A;
  defpred P[object] means ex A st $1 in Rank A;
  consider Power being set such that
A3: for x being object holds x in Power iff x in Tarski-Class X & P[x]
from XBOOLE_0:sch 1;
  defpred P[object,object] means
  ex A st $2 = A & not $1 in Rank A & $1 in Rank succ A;
A4: for x,y,z being object st P[x,y] & P[x,z] holds y = z
  proof
    let x,y,z be object;
    given A1 being Ordinal such that
A5: y = A1 and
A6: not x in Rank A1 and
A7: x in Rank succ A1;
    given A2 being Ordinal such that
A8: z = A2 and
A9: not x in Rank A2 and
A10: x in Rank succ A2;
    assume y <> z;
then
A11: A1 in A2 or A2 in A1 by A5,A8,ORDINAL1:14;
 now
      assume succ A1 c= A2;
then   Rank succ A1 c= Rank A2 by Th37;
      hence contradiction by A7,A9;
    end;
then  Rank succ A2 c= Rank A1 by A11,Th37,ORDINAL1:21;
    hence contradiction by A6,A10;
  end;
  consider Y such that
A12: for x being object holds x in Y iff
  ex y being object st y in Power & P[y,x] from TARSKI:sch 1(A4);
 now
    let A;
 Rank A /\ Tarski-Class X <> Rank succ A /\ Tarski-Class X by A1,A2,Th46;
    then consider y being object such that
    A13: not
 (y in Rank A /\ Tarski-Class X iff y in Rank succ A /\ Tarski-Class X)
    by TARSKI:2;
A14:  A c= succ A by ORDINAL1:6,def 2;
then
A15: Rank A c= Rank succ A by Th37;
  Rank A /\ Tarski-Class X c= Rank succ A /\ Tarski-Class X
      by XBOOLE_1:26,A14,Th37;
then A16: y in Rank succ A by A13,XBOOLE_0:def 4;
A17: y in Tarski-Class X by A13,XBOOLE_0:def 4;
   then
A18: not y in Rank A by A13,A15,XBOOLE_0:def 4;
 y in Power by A3,A16,A17;
    hence A in Y by A12,A16,A18;
  end;
  hence contradiction by ORDINAL1:26;
end;
