In [1]:

```
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
```

In [2]:

```
%begin axioms
```

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

In [3]:

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

In [4]:

```
%end axioms
```