let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
All ((All (('not' a),A,G)),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
let a be Function of Y,BOOLEAN; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
All ((All (('not' a),A,G)),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y st G is independent holds
All ((All (('not' a),A,G)),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
let A, B be a_partition of Y; :: thesis: ( G is independent implies All ((All (('not' a),A,G)),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G)) )
A1: ( 'not' (Ex ((Ex (a,A,G)),B,G)) = All (('not' (Ex (a,A,G))),B,G) & All (('not' (Ex (a,A,G))),B,G) = All ((All (('not' a),A,G)),B,G) ) by BVFUNC_2:19;
assume G is independent ; :: thesis: All ((All (('not' a),A,G)),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
hence All ((All (('not' a),A,G)),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G)) by A1, Th33; :: thesis: verum