reserve x,A,B,X,X9,Y,Y9,Z,V for set;

theorem Th83:
  X misses Y iff X \ Y = X
proof
  thus X misses Y implies X \ Y = X
  proof
    assume
A1: X /\ Y = {};
    thus for x being object holds x in X \ Y implies x in X by XBOOLE_0:def 5;
    let x be object;
    not x in X /\ Y implies not x in X or not x in Y by XBOOLE_0:def 4;
    hence thesis by A1,XBOOLE_0:def 5;
  end;
  assume
A2: X \ Y = X;
  not ex x being object st x in X /\ Y
  proof
    given x being object such that
A3: x in X /\ Y;
    x in X & x in Y by A3,XBOOLE_0:def 4;
    hence contradiction by A2,XBOOLE_0:def 5;
  end;
  hence thesis by XBOOLE_0:4;
end;
