%   ORIGINAL: 'h4/thm/rich_list/LIST_TO_SET_EQ_SING_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !x ls. 'h4/const/list/LIST_TO_SET' ls = 'h4/const/pred_set/INSERT' x 'h4/const/pred_set/EMPTY' <=> ~(ls = 'h4/const/list/NIL') /\ 'h4/const/list/EVERY' ($equals x) ls
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !x ls. 'h4/const/list/LIST_TO_SET' ls = 'h4/const/pred_set/INSERT' x 'h4/const/pred_set/EMPTY' <=> ~(ls = 'h4/const/list/NIL') /\ 'h4/const/list/EVERY' ($equals x) ls
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f57686,V_3f57682]: ![F, G]: (![X]: s(V_3f57682,happ(s(fun(V_3f57686,V_3f57682),F),s(V_3f57686,X))) = s(V_3f57682,happ(s(fun(V_3f57686,V_3f57682),G),s(V_3f57686,X))) => s(fun(V_3f57686,V_3f57682),F) = s(fun(V_3f57686,V_3f57682),G))).
fof('h4/thm/rich_list/LIST_TO_SET_EQ_SING_', conjecture, ![A]: ![X, Ls]: (s(fun(A,bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'(A),Ls))) = s(fun(A,bool),'h4/const/pred_set/INSERT'(s(A,X),s(fun(A,bool),'h4/const/pred_set/EMPTY'))) <=> (~ (s('h4/type/list/list'(A),Ls) = s('h4/type/list/list'(A),'h4/const/list/NIL')) & p(s(bool,'h4/const/list/EVERY'(s(fun(A,bool),'$equals'(s(A,X))),s('h4/type/list/list'(A),Ls))))))).
