let R be Relation; :: thesis: ( R is irreflexive & R is transitive implies R is asymmetric )
assume that
A1: R is_irreflexive_in field R and
A2: R is_transitive_in field R ; :: according to RELAT_2:def 10,RELAT_2:def 16 :: thesis: R is asymmetric
let a be object ; :: according to RELAT_2:def 5,RELAT_2:def 13 :: thesis: for y being object st a in field R & y in field R & [a,y] in R holds
not [y,a] in R
let b be object ; :: thesis: ( a in field R & b in field R & [a,b] in R implies not [b,a] in R )
assume that
A3: a in field R and
A4: b in field R ; :: thesis: ( not [a,b] in R or not [b,a] in R )
not [a,a] in R by A1, A3;
hence ( not [a,b] in R or not [b,a] in R ) by A2, A3, A4; :: thesis: verum