let R be Relation; :: thesis: for x being set st x is_inferior_of R & R is antisymmetric holds
x is_minimal_in R
let x be set ; :: thesis: ( x is_inferior_of R & R is antisymmetric implies x is_minimal_in R )
assume that
A1: x is_inferior_of R and
A2: R is antisymmetric ; :: thesis: x is_minimal_in R
thus A3: x in field R by A1, Def14; :: according to ORDERS_1:def 12 :: thesis: for y being set holds
( not y in field R or not y <> x or not [y,x] in R )
let y be set ; :: thesis: ( not y in field R or not y <> x or not [y,x] in R )
assume that
A4: y in field R and
A5: y <> x and
A6: [y,x] in R ; :: thesis: contradiction
( [x,y] in R & R is_antisymmetric_in field R ) by A1, A2, A4, A5, Def14, RELAT_2:def 12;
hence contradiction by A3, A4, A5, A6, RELAT_2:def 4; :: thesis: verum