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

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

end;

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