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 Th46:
  X is epsilon-transitive &
  Rank A /\ Tarski-Class X = Rank succ A /\ Tarski-Class X implies
  Tarski-Class X c= Rank A
proof
  assume that
A1: X is epsilon-transitive and
A2: Rank A /\ Tarski-Class X = Rank succ A /\ Tarski-Class X;
  given x being object such that
A3: x in Tarski-Class X & not x in Rank A;
 x in (Tarski-Class X) \ Rank A by A3,XBOOLE_0:def 5;
  then consider Y such that
A4: Y in (Tarski-Class X) \ Rank A and
A5: not ex x be object st x in (Tarski-Class X) \ Rank A & x in Y by TARSKI:3;
A6: Y c= Tarski-Class X by A4,ORDINAL1:def 2,A1,Th23;
 Y c= Rank A
  proof
    let x be object;
    assume
A7: x in Y;
then  not x in (Tarski-Class X) \ Rank A by A5;
    hence thesis by A6,A7,XBOOLE_0:def 5;
  end;
then  Y in Rank succ A by Th32;
then A8: Y in Rank succ A /\ Tarski-Class X by A4,XBOOLE_0:def 4;
 not Y in Rank A by A4,XBOOLE_0:def 5;
  hence contradiction by A2,A8,XBOOLE_0:def 4;
end;
