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, Or, Equals, InSet, Union
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_any_A, general_unionall_Ryn
from proveit.numbers import NaturalPos
%begin axioms
```

**Membership in the union of sets is the same as having membership in any of the sets being unified:**

In [2]:

```
union_def = Forall(m, Forall((x, A_1_to_m),
Equals(InSet(x, Union(A_1_to_m)), x_in_any_A).with_wrap_after_operator()),
domain=NaturalPos)
```

In [3]:

```
union_all_def = Forall(
n, Forall((S_1_to_n, Q, R, x),
Equals(InSet(x, general_unionall_Ryn),
Exists(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()),
domain=NaturalPos)
```

In [4]:

```
%end axioms
```