Proof of proveit.logic.sets.enumeration.in_singleton_eval_false theorem

see dependencies

import proveit
from proveit import defaults
from proveit import x, y
from proveit.logic import NotEquals, Not, Equals, TRUE, FALSE, InSet, NotInSet, Set
from proveit.logic.equality  import not_equals_def
from proveit.logic.equality import sub_right_side_into
from proveit.logic.sets.enumeration import not_in_singleton_equiv, in_singleton_eval_true, singleton_def
%proving in_singleton_eval_false

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

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

defaults.assumptions = in_singleton_eval_false.conditions

defaults.assumptions:

step1 = not_equals_def.instantiate({x:x, y:y})

step1:  ⊢  

state1 = Equals(NotEquals(x,y), TRUE).prove()

state1:  ⊢  

step1.sub_right_side_into(state1)

 ⊢  

state2 = Equals(Equals(x,y), FALSE).prove()

state2:  ⊢  

state3 = singleton_def.instantiate({x:x, y:y})

state3:  ⊢  

goal = state3.sub_left_side_into(state2)

goal:  ⊢  

goal.generalize([x,y], conditions = in_singleton_eval_false.conditions)

 ⊢  

%qed

proveit.logic.sets.enumeration.in_singleton_eval_false 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	axiom		⊢
	proveit.logic.booleans.negation.negation_elim
6	instantiation	8, 9	⊢
	: , :
7	conjecture		⊢
	proveit.logic.sets.enumeration.singleton_def
8	theorem		⊢
	proveit.logic.equality.unfold_not_equals
9	assumption		⊢