let V be non empty set ; :: thesis: for a being Object of (Ens V) st a = {} holds
a is initial
let a be Object of (Ens V); :: thesis: ( a = {} implies a is initial )
assume A1: a = {} ; :: thesis: a is initial
let b be Object of (Ens V); :: according to CAT_1:def 19 :: thesis: ( not Hom (a,b) = {} & ex b₁ being Morphism of a,b st
for b₂ being Morphism of a,b holds b₁ = b₂ )
Maps ((@ a),(@ b)) <> {} by A1, Th15;
hence A2: Hom (a,b) <> {} by Th26; :: thesis: ex b₁ being Morphism of a,b st
for b₂ being Morphism of a,b holds b₁ = b₂
set m = [[(@ a),(@ b)],{}];
{} is Function of (@ a),(@ b) by A1, RELSET_1:12;
then [[(@ a),(@ b)],{}] in Maps ((@ a),(@ b)) by A1, Th15;
then [[(@ a),(@ b)],{}] in Hom (a,b) by Th26;
then reconsider f = [[(@ a),(@ b)],{}] as Morphism of a,b by CAT_1:def 5;
take f ; :: thesis: for b₁ being Morphism of a,b holds f = b₁
let g be Morphism of a,b; :: thesis: f = g
reconsider m9 = g as Element of Maps V ;
g in Hom (a,b) by A2, CAT_1:def 5;
then A3: g in Maps ((@ a),(@ b)) by Th26;
then A4: m9 = [[(@ a),(@ b)],(m9 `2)] by Th16;
then m9 `2 is Function of (@ a),(@ b) by A3, Lm4;
hence f = g by A1, A4; :: thesis: verum