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
A3: R is_antisymmetric_in field R by A2;
thus A4: x in field R by A1; :: according to ORDERS_1:def 13 :: 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
A5: y in field R and
A6: y <> x and
A7: [y,x] in R ; :: thesis: contradiction
[x,y] in R by A1, A5, A6;
hence contradiction by A4, A5, A6, A7, A3; :: thesis: verum