thus ( X misses Y implies for n being Nat holds
( not n in X or not n in Y ) ) by XBOOLE_0:3; :: thesis: ( ( for n being Nat holds
( not n in X or not n in Y ) ) implies not X meets Y )
assume A1: for n being Nat holds
( not n in X or not n in Y ) ; :: thesis: not X meets Y
assume X meets Y ; :: thesis: contradiction
then ex x being set st
( x in X & x in Y ) by XBOOLE_0:3;
hence contradiction by A1; :: thesis: verum