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.logic import Forall, Exists, And, Equals, InSet, Intersect, IntersectAll
from proveit import m, n, x, Q, R
from proveit.core_expr_types import y_1_to_n, A_1_to_m, Q__y_1_to_n, R__y_1_to_n, S_1_to_n
from proveit.logic.sets import x_in_every_A, general_intersectall_Ryn
from proveit.numbers import NaturalPos
%begin axioms
```

**Membership in the intersection of sets is the same as having membership in all of the sets being intersected:**

In [2]:

```
intersection_def = Forall(m, Forall((x, A_1_to_m),
Equals(InSet(x, Intersect(A_1_to_m)),
x_in_every_A).with_wrap_after_operator()),
domain=NaturalPos)
```

**We can intersect over all instances of sets under some criteria, but there must be at least one set (we cannot allow a universal set -- we cannot allow a set to contain itself or we will be subject to Russel's paradox):**

In [3]:

```
intersect_all_def = Forall(n,
Forall((S_1_to_n, Q, R, x),
Equals(InSet(x, general_intersectall_Ryn),
Forall(y_1_to_n, InSet(x, R__y_1_to_n),
domains=[S_1_to_n], conditions=[Q__y_1_to_n]))
.with_wrap_before_operator(),
condition=Exists(y_1_to_n, Q__y_1_to_n,domains=[S_1_to_n])),
domain=NaturalPos)
```

In [4]:

```
%end axioms
```