let X be set ; :: thesis: X \ {} = X
thus X \ {} c= X :: according to XBOOLE_0:def 10 :: thesis: X c= X \ {}

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X \ {} or x in X )
assume x in X \ {} ; :: thesis: x in X
hence x in X by XBOOLE_0:def 5; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in X \ {} )
A1: not x in {} by XBOOLE_0:def 1;
assume x in X ; :: thesis: x in X \ {}
hence x in X \ {} by A1, XBOOLE_0:def 5; :: thesis: verum