let X, S be non empty set ; :: thesis: for R being Relation of X st R is irreflexive & field R c= S holds
R is_irreflexive_in S
let R be Relation of X; :: thesis: ( R is irreflexive & field R c= S implies R is_irreflexive_in S )
assume that
A1: R is irreflexive and
A2: field R c= S ; :: thesis: R is_irreflexive_in S
let x be set ; :: according to RELAT_2:def 2 :: thesis: ( not x in S or not [x,x] in R )
S = (field R) \/ (S \ (field R)) by A2, XBOOLE_1:45;
then A3: ( not x in S or x in field R or x in S \ (field R) ) by XBOOLE_0:def 3;
A4: ( x in S \ (field R) implies not [x,x] in R )

proof

assume x in S \ (field R) ; :: thesis: not [x,x] in R
then x in S \ ((dom R) \/ (rng R)) by RELAT_1:def 6;
then x in (S \ (dom R)) /\ (S \ (rng R)) by XBOOLE_1:53;
then x in S \ (rng R) by XBOOLE_0:def 4;
then not x in rng R by XBOOLE_0:def 5;
hence not [x,x] in R by RELAT_1:def 5; :: thesis: verum

end;

assume A5: x in S ; :: thesis: not [x,x] in R
R is_irreflexive_in field R by A1, RELAT_2:def 10;
hence not [x,x] in R by A5, A3, A4, RELAT_2:def 2; :: thesis: verum