let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN st a 'imp' b = I_el Y & a 'imp' ('not' b) = I_el Y holds
'not' a = I_el Y
let a, b be Function of Y,BOOLEAN; :: thesis: ( a 'imp' b = I_el Y & a 'imp' ('not' b) = I_el Y implies 'not' a = I_el Y )
assume that
A1: a 'imp' b = I_el Y and
A2: a 'imp' ('not' b) = I_el Y ; :: thesis: 'not' a = I_el Y
for x being Element of Y holds ('not' a) . x = TRUE
proof
let x be Element of Y; :: thesis: ('not' a) . x = TRUE
(a 'imp' b) . x = TRUE by A1, BVFUNC_1:def 11;
then A3: ('not' (a . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 8;
(a 'imp' ('not' b)) . x = TRUE by A2, BVFUNC_1:def 11;
then A4: ('not' (a . x)) 'or' (('not' b) . x) = TRUE by BVFUNC_1:def 8;
A5: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;
now :: thesis: ( ( 'not' (a . x) = TRUE & 'not' (a . x) = TRUE & ('not' a) . x = TRUE ) or ( 'not' (a . x) = TRUE & ('not' b) . x = TRUE & ('not' a) . x = TRUE ) or ( b . x = TRUE & 'not' (a . x) = TRUE & ('not' a) . x = TRUE ) or ( b . x = TRUE & ('not' b) . x = TRUE & ('not' a) . x = TRUE ) )
per cases ( ( 'not' (a . x) = TRUE & 'not' (a . x) = TRUE ) or ( 'not' (a . x) = TRUE & ('not' b) . x = TRUE ) or ( b . x = TRUE & 'not' (a . x) = TRUE ) or ( b . x = TRUE & ('not' b) . x = TRUE ) ) by A3, A5, A4, BINARITH:3;
case ( 'not' (a . x) = TRUE & 'not' (a . x) = TRUE ) ; :: thesis: ('not' a) . x = TRUE
hence ('not' a) . x = TRUE by MARGREL1:def 19; :: thesis: verum
end;
case ( 'not' (a . x) = TRUE & ('not' b) . x = TRUE ) ; :: thesis: ('not' a) . x = TRUE
hence ('not' a) . x = TRUE by MARGREL1:def 19; :: thesis: verum
end;
end;
end;
hence ('not' a) . x = TRUE ; :: thesis: verum
end;
hence 'not' a = I_el Y by BVFUNC_1:def 11; :: thesis: verum