reserve x,y,z for Variable,
  H for ZF-formula,
  E for non empty set,
  a,b,c,X,Y,Z for set,
  u,v,w for Element of E,
  f,g,h,i,j for Function of VAR,E;

theorem
  E is epsilon-transitive implies (E |= the_axiom_of_infinity iff ex X
  st X in E & X <> {} & for Y st Y in X ex Z st Y c< Z & Z in X)
proof
  assume
A1: E is epsilon-transitive;
  thus E |= the_axiom_of_infinity implies ex X st X in E & X <> {} & for Y st
  Y in X ex Z st Y c< Z & Z in X
  proof
    assume E |= the_axiom_of_infinity;
    then consider u such that
A2: u <> {} and
A3: for v st v in u ex w st v c< w & w in u by A1,Th6;
    reconsider X = u as set;
    take X;
    thus X in E & X <> {} by A2;
    let Y such that
A4: Y in X;
    X c= E by A1;
    then reconsider v = Y as Element of E by A4;
    consider w such that
A5: v c< w & w in u by A3,A4;
    reconsider w as set;
    take w;
    thus thesis by A5;
  end;
  given X such that
A6: X in E and
A7: X <> {} and
A8: for Y st Y in X ex Z st Y c< Z & Z in X;
  ex u st u <> {} & for v st v in u ex w st v c< w & w in u
  proof
    reconsider u = X as Element of E by A6;
    take u;
    thus u <> {} by A7;
    let v;
    assume v in u;
    then consider Z such that
A9: v c< Z and
A10: Z in X by A8;
    X c= E by A1,A6;
    then reconsider w = Z as Element of E by A10;
    take w;
    thus thesis by A9,A10;
  end;
  hence thesis by A1,Th6;
end;
