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

theorem Th30:
  for A,B st for x being Element of E holds x in A iff not x in B holds A = B`
proof
  let A,B be Subset of E;
  assume
A1: for x being Element of E holds x in A iff not x in B;
  thus A c= B`
  proof
    let x be object;
    assume
A2: x in A;
  reconsider x as set by TARSKI:1;
    A in bool E by Def1;
    then A c= E by ZFMISC_1:def 1;
    then x in E by A2;
    then x is Element of E by Def1;
    then
A3: not x in B by A1,A2;
    x in E by A2,Lm1;
    hence thesis by A3,XBOOLE_0:def 5;
  end;
  let x be object;
  assume
A4: x in B`;
  reconsider x as set by TARSKI:1;
  B` in bool E by Def1;
  then B` c= E by ZFMISC_1:def 1;
  then x in E by A4;
  then reconsider x as Element of E by Def1;
  not x in B by A4,XBOOLE_0:def 5;
  hence thesis by A1;
end;
