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

let x be object ; :: thesis: ( x in {1,2,3} /\ {4} iff contradiction )
( x in {1,2,3} /\ {4} iff ( x in {1,2,3} & x in {4} ) ) by XBOOLE_0:def 4;
then ( x in {1,2,3} /\ {4} iff ( ( x = 1 or x = 2 or x = 3 ) & x = 4 ) ) by TARSKI:def 1, ENUMSET1:def 1;
hence ( x in {1,2,3} /\ {4} iff contradiction ) ; :: thesis: verum

end;

hence {1,2,3} /\ {4} = {} by XBOOLE_0:def 1; :: thesis: verum