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

theorem
  A misses B & A` misses B` implies A = B`
proof
  assume that
A1: A misses B and
A2: A` misses B`;
A3: A in bool E by Def1;
  thus A c= B`
  proof
    let x be object;
    assume
A4: x in A;
    then
A5: not x in B by A1,XBOOLE_0:3;
    A c= E by A3,ZFMISC_1:def 1;
    then x in E by A4;
    hence thesis by A5,XBOOLE_0:def 5;
  end;
  let x be object;
A6: x in A` implies not x in B` by A2,XBOOLE_0:3;
  assume
A7: x in B`;
  then x in E by XBOOLE_0:def 5;
  hence thesis by A7,A6,XBOOLE_0:def 5;
end;
