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 holds 'not' (Ex (Ex a,A,G),B,G) '<' Ex ('not' (Ex 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 holds 'not' (Ex (Ex a,A,G),B,G) '<' Ex ('not' (Ex a,B,G)),A,G
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y holds 'not' (Ex (Ex a,A,G),B,G) '<' Ex ('not' (Ex a,B,G)),A,G
let A, B be a_partition of Y; :: thesis: 'not' (Ex (Ex a,A,G),B,G) '<' Ex ('not' (Ex a,B,G)),A,G
( 'not' (Ex (Ex a,A,G),B,G) '<' 'not' (All (Ex a,B,G),A,G) & Ex ('not' (Ex a,B,G)),A,G = Ex (All ('not' a),B,G),A,G ) by Th11, BVFUNC_2:21;
hence 'not' (Ex (Ex a,A,G),B,G) '<' Ex ('not' (Ex a,B,G)),A,G by BVFUNC11:21; :: thesis: verum