reserve E,X,Y,x for set;
reserve A,B,C for Subset of E;

theorem
  (for x being Element of E holds x in A) implies E = A
proof
  assume
A1: for x being Element of E holds x in A;
  thus E c= A
  proof
    let a be object;
    assume
A2:   a in E;
  reconsider a as set by TARSKI:1;
     a is Element of E by Def1,A2;
    hence thesis by A1;
  end;
  A in bool E by Def1;
  hence thesis by ZFMISC_1:def 1;
end;
