Proof of proveit.logic.sets.membership.not_in_set_is_bool theorem

import proveit
from proveit import defaults
from proveit import x, A, S
from proveit.logic import InSet
from proveit.logic.booleans.negation import closure
from proveit.logic.sets.membership  import not_in_set_def
theory = proveit.Theory() # the theorem's theory

%proving not_in_set_is_bool

With these allowed/disallowed theorem/theory presumptions (e.g., to avoid circular dependencies), we begin our proof of
not_in_set_is_bool:
(see dependencies)

defaults.assumptions = not_in_set_is_bool.all_conditions()

defaults.assumptions:

closure

 ⊢  

closure_spec = closure.instantiate({A:InSet(x, S)})

closure_spec:  ⊢  

not_in_set_def

 ⊢  

not_in_set_def_spec = not_in_set_def.instantiate({x:x, S:S})

not_in_set_def_spec:  ⊢  

not_in_set_def_spec.sub_left_side_into(closure_spec)

 ⊢  

not_in_set_is_bool may now be readily provable (assuming required theorems are usable).  Simply execute "%qed".

%qed

proveit.logic.sets.membership.not_in_set_is_bool has been proven.

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	2, 3, 4	⊢
	: , : , :
2	theorem		⊢
	proveit.logic.equality.sub_left_side_into
3	instantiation	5, 6	⊢
	:
4	instantiation	7	⊢
	: , :
5	conjecture		⊢
	proveit.logic.booleans.negation.closure
6	assumption		⊢
7	axiom		⊢
	proveit.logic.sets.membership.not_in_set_def