let X, Y be set ; :: thesis: ( X misses Y iff X \ Y = X )
thus ( X misses Y implies X \ Y = X ) :: thesis: ( X \ Y = X implies X misses Y )

proof

assume A1: X /\ Y = {} ; :: according to XBOOLE_0:def 7 :: thesis: X \ Y = X
thus for x being object st x in X \ Y holds
x in X by XBOOLE_0:def 5; :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: X c= X \ Y
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in X \ Y )
( x in X /\ Y or not x in X or not x in Y ) by XBOOLE_0:def 4;
hence ( not x in X or x in X \ Y ) by A1, XBOOLE_0:def 5; :: thesis: verum

end;

assume A2: X \ Y = X ; :: thesis: X misses Y
for x being object holds not x in X /\ Y

proof

given x being object such that A3: x in X /\ Y ; :: thesis: contradiction
( x in X & x in Y ) by A3, XBOOLE_0:def 4;
hence contradiction by A2, XBOOLE_0:def 5; :: thesis: verum

end;

hence X misses Y by XBOOLE_0:4; :: thesis: verum