let V be non empty set ; :: thesis: for a being Object of (Ens V) st ex x being set st a = {x} holds
a is terminal
let a be Object of (Ens V); :: thesis: ( ex x being set st a = {x} implies a is terminal )
given x being set such that A1: a = {x} ; :: thesis: a is terminal
let b be Object of (Ens V); :: according to CAT_1:def 18 :: thesis: ( not Hom (b,a) = {} & ex b₁ being Morphism of b,a st
for b₂ being Morphism of b,a holds b₁ = b₂ )
set h = the Function of (@ b),(@ a);
set m = [[(@ b),(@ a)], the Function of (@ b),(@ a)];
A2: [[(@ b),(@ a)], the Function of (@ b),(@ a)] in Maps ((@ b),(@ a)) by A1, Th15;
hence A3: Hom (b,a) <> {} by Th26; :: thesis: ex b₁ being Morphism of b,a st
for b₂ being Morphism of b,a holds b₁ = b₂
[[(@ b),(@ a)], the Function of (@ b),(@ a)] in Hom (b,a) by A2, Th26;
then reconsider f = [[(@ b),(@ a)], the Function of (@ b),(@ a)] as Morphism of b,a by CAT_1:def 5;
take f ; :: thesis: for b₁ being Morphism of b,a holds f = b₁
let g be Morphism of b,a; :: thesis: f = g
reconsider m9 = g as Element of Maps V ;
g in Hom (b,a) by A3, CAT_1:def 5;
then A4: g in Maps ((@ b),(@ a)) by Th26;
then A5: m9 = [[(@ b),(@ a)],(m9 `2)] by Th16;
then m9 `2 is Function of (@ b),(@ a) by A4, Lm4;
hence f = g by A1, A5, FUNCT_2:51; :: thesis: verum