now :: thesis: for x being object holds
( x in {4} \ {1,2,3} iff x in {4} )

let x be object ; :: thesis: ( x in {4} \ {1,2,3} iff x in {4} )
( x in {4} \ {1,2,3} iff ( x in {4} & not x in {1,2,3} ) ) by XBOOLE_0:def 5;
then ( x in {4} \ {1,2,3} iff ( x = 4 & not x = 1 & not x = 2 & not x = 3 ) ) by TARSKI:def 1, ENUMSET1:def 1;
hence ( x in {4} \ {1,2,3} iff x in {4} ) by TARSKI:def 1; :: thesis: verum

end;

hence {4} \ {1,2,3} = {4} by TARSKI:2; :: thesis: verum