let X, Y be set ; :: thesis: ( X c= Y \ X implies X = {} )
assume A1: X c= Y \ X ; :: thesis: X = {}
thus X c= {} :: according to XBOOLE_0:def 10 :: thesis: {} c= X

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in {} )
assume A2: x in X ; :: thesis: x in {}
then x in Y \ X by A1;
hence x in {} by A2, XBOOLE_0:def 5; :: thesis: verum

end;

thus {} c= X ; :: thesis: verum