let U be set ; :: thesis: for A, B being Subset of U st not A c= B holds
Inter (A,B) = {}
let A, B be Subset of U; :: thesis: ( not A c= B implies Inter (A,B) = {} )
assume A1: not A c= B ; :: thesis: Inter (A,B) = {}
assume Inter (A,B) <> {} ; :: thesis: contradiction
then consider x being object such that
A2: x in Inter (A,B) by XBOOLE_0:def 1;
reconsider x = x as set by TARSKI:1;
( A c= x & x c= B ) by A2, Th1;
hence contradiction by A1; :: thesis: verum