let X be BCI-algebra; :: thesis: for IT being non empty Subset of X st IT is Ideal of X holds
for x, y being Element of X st x in IT & y <= x holds
y in IT
let IT be non empty Subset of X; :: thesis: ( IT is Ideal of X implies for x, y being Element of X st x in IT & y <= x holds
y in IT )
assume A1: IT is Ideal of X ; :: thesis: for x, y being Element of X st x in IT & y <= x holds
y in IT
let x, y be Element of X; :: thesis: ( x in IT & y <= x implies y in IT )
assume that
A2: x in IT and
A3: y <= x ; :: thesis: y in IT
y \ (0. X) <= x by A3, Th2;
then A4: y in { z where z is Element of X : z \ (0. X) <= x } ;
0. X is Element of IT by A1, Def18;
then { z where z is Element of X : z \ (0. X) <= x } c= IT by A1, A2, Lm5;
hence y in IT by A4; :: thesis: verum