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;
reserve n for Element of omega;

theorem Th53:
  X c= Y & Y is epsilon-transitive implies the_transitive-closure_of X c= Y
proof
  assume that
A1: X c= Y and
A2: Y is epsilon-transitive;
  let x be object;
  assume x in the_transitive-closure_of X;
  then consider f,n such that
A3: x in f.n and dom f = omega and
A4: f.0 = X and
A5: for k being Nat holds f.(succ k) = union(f.k) by Def7;
  defpred P[Nat] means f.$1 c= Y;
A6: P[0] by A1,A4;
A7: for k being Nat st P[k] holds P[succ k]
  proof
    let k be Nat;
    assume f.k c= Y;
then A8: union (f.k) c= union Y by ZFMISC_1:77;
 f.(succ k) = union (f.k) & union Y c= Y by A2,A5,Th48;
    hence thesis by A8,XBOOLE_1:1;
  end;
 P[n] from ORDINAL2:sch 17(A6,A7);
  hence thesis by A3;
end;
