let U be set ; :: thesis: for A, B being Subset of U st Inter (A,B) <> {} holds

A c= B

let A, B be Subset of U; :: thesis: ( Inter (A,B) <> {} implies A c= B )

assume Inter (A,B) <> {} ; :: thesis: A c= B

then consider x being object such that

A1: 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 A1, Th1;

hence A c= B ; :: thesis: verum

A c= B

let A, B be Subset of U; :: thesis: ( Inter (A,B) <> {} implies A c= B )

assume Inter (A,B) <> {} ; :: thesis: A c= B

then consider x being object such that

A1: 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 A1, Th1;

hence A c= B ; :: thesis: verum