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

see dependencies

In [1]:
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".

In [2]:
defaults.assumptions = in_singleton_eval_false.conditions

defaults.assumptions:
In [3]:
step1 = not_equals_def.instantiate({x:x, y:y})

step1:
In [4]:
state1 = Equals(NotEquals(x,y), TRUE).prove()

state1:
In [5]:
step1.sub_right_side_into(state1)

In [6]:
state2 = Equals(Equals(x,y), FALSE).prove()

state2:
In [7]:
state3 = singleton_def.instantiate({x:x, y:y})

state3:
In [8]:
goal = state3.sub_left_side_into(state2)

goal:
In [9]:
goal.generalize([x,y], conditions = in_singleton_eval_false.conditions)

In [10]:
%qed

proveit.logic.sets.enumeration.in_singleton_eval_false has been proven.

Out[10]:
step typerequirementsstatement
0generalization1
1instantiation2, 3, 4
: , : , :
2theorem
proveit.logic.equality.sub_left_side_into
3instantiation5, 6
:
4instantiation7
: , :
5axiom
proveit.logic.booleans.negation.negation_elim
6instantiation8, 9
: , :
7conjecture
proveit.logic.sets.enumeration.singleton_def
8theorem
proveit.logic.equality.unfold_not_equals
9assumption