%   ORIGINAL: 'h4/thm/list/mem_exists_set_'
% 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 y l. 'h4/const/bool/IN' ('h4/const/pair/,' x y) ('h4/const/list/LIST_TO_SET' l) ==> (?z. x = 'h4/const/pair/FST' z /\ 'h4/const/bool/IN' z ('h4/const/list/LIST_TO_SET' l))
%   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 y l. 'h4/const/bool/IN' ('h4/const/pair/,' x y) ('h4/const/list/LIST_TO_SET' l) ==> (?z. x = 'h4/const/pair/FST' z /\ 'h4/const/bool/IN' z ('h4/const/list/LIST_TO_SET' l))
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_3f12494,V_3f12490]: ![F, G]: (![X]: s(V_3f12490,happ(s(fun(V_3f12494,V_3f12490),F),s(V_3f12494,X))) = s(V_3f12490,happ(s(fun(V_3f12494,V_3f12490),G),s(V_3f12494,X))) => s(fun(V_3f12494,V_3f12490),F) = s(fun(V_3f12494,V_3f12490),G))).
fof('h4/thm/list/mem_exists_set_', conjecture, ![A,B]: ![X, Y, L]: (p(s(bool,'h4/const/bool/IN'(s('h4/type/pair/prod'(A,B),'h4/const/pair/,'(s(A,X),s(B,Y))),s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'('h4/type/pair/prod'(A,B)),L)))))) => ?[Z]: (s(A,X) = s(A,'h4/const/pair/FST'(s('h4/type/pair/prod'(A,B),Z))) & p(s(bool,'h4/const/bool/IN'(s('h4/type/pair/prod'(A,B),Z),s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'('h4/type/pair/prod'(A,B)),L))))))))).
