let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
'not' (All (Ex a,A,G),B,G) '<' 'not' (All (All a,B,G),A,G)
let a be Element of Funcs Y,BOOLEAN ; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
'not' (All (Ex a,A,G),B,G) '<' 'not' (All (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
'not' (All (Ex a,A,G),B,G) '<' 'not' (All (All a,B,G),A,G)
let A, B be a_partition of Y; :: thesis: ( G is independent implies 'not' (All (Ex a,A,G),B,G) '<' 'not' (All (All a,B,G),A,G) )
assume G is independent ; :: thesis: 'not' (All (Ex a,A,G),B,G) '<' 'not' (All (All a,B,G),A,G)
then A1: Ex ('not' (Ex a,A,G)),B,G '<' 'not' (All (All a,B,G),A,G) by Th7;
Ex ('not' (Ex a,A,G)),B,G = Ex (All ('not' a),A,G),B,G by BVFUNC_2:21;
hence 'not' (All (Ex a,A,G),B,G) '<' 'not' (All (All a,B,G),A,G) by A1, BVFUNC11:21; :: thesis: verum