let R be Relation; :: thesis: ( R is irreflexive & R is transitive implies R is asymmetric )
assume that
A4: R is_irreflexive_in field R and
A3: R is_transitive_in field R ; :: according to RELAT_2:def 10,RELAT_2:def 16 :: thesis: R is asymmetric
let a be set ; :: according to RELAT_2:def 5,RELAT_2:def 13 :: thesis: for y being set st a in field R & y in field R & [a,y] in R holds
not [y,a] in R
let b be set ; :: thesis: ( a in field R & b in field R & [a,b] in R implies not [b,a] in R )
assume that
A5: a in field R and
A6: b in field R ; :: thesis: ( not [a,b] in R or not [b,a] in R )
not [a,a] in R by A4, A5, Def2;
hence ( not [a,b] in R or not [b,a] in R ) by A3, A5, A6, Def8; :: thesis: verum