Axioms for the theory of proveit.logic.classes.membership

import proveit
# Prepare this notebook for defining the axioms of a theory:
%axioms_notebook # Keep this at the top following 'import proveit'.

from proveit import x, C
from proveit.logic import Forall, Not, Equals, InClass, NotInClass

%begin axioms

Defining axioms for theory 'proveit.logic.classes.membership'
Subsequent end-of-cell assignments will define axioms
%end_axioms will finalize the definitions

$x \notin C$ is defined as the negation of $x \in C$ for all $x$ and $C$:

not_in_class_def = Forall((x, C), Equals(NotInClass(x, C), 
                                         Not(InClass(x, C))))

not_in_class_def:

%end axioms

These axioms may now be imported from the theory package: proveit.logic.classes.membership