Slate - Union is associative

Proposition 1.1.17. Let U be a set, R, S, T ⊆ U. Then:

(R ∪ S) ∪ T = R ∪ (S ∪ T)

Proof.

Let x ∈ (R ∪ S) ∪ T. Then x ∈ R ∪ (S ∪ T):
x ∈ (R ∪ S) ∪ T ⇒^def x ∈ R ∪ S or x ∈ T
- Assume x ∈ R ∪ S. Then x ∈ R ∪ (S ∪ T):
  x ∈ R ∪ S ⇒^def x ∈ R or x ∈ S
  - Assume x ∈ R. Then x ∈ R ∪ (S ∪ T) by definition.
  - Assume x ∈ S. Then x ∈ R ∪ (S ∪ T):
    We have x ∈ S ∪ T by definition.
- Assume x ∈ T. Then x ∈ R ∪ (S ∪ T):
  We have x ∈ S ∪ T by definition.
Let x ∈ R ∪ (S ∪ T). Then x ∈ (R ∪ S) ∪ T:
x ∈ R ∪ (S ∪ T) ⇒^def x ∈ R or x ∈ S ∪ T
- Assume x ∈ R. Then x ∈ (R ∪ S) ∪ T:
  We have x ∈ R ∪ S by definition.
- Assume x ∈ S ∪ T. Then x ∈ (R ∪ S) ∪ T:
  x ∈ S ∪ T ⇒^def x ∈ S or x ∈ T
  - Assume x ∈ S. Then x ∈ (R ∪ S) ∪ T:
    We have x ∈ R ∪ S by definition.
  - Assume x ∈ T. Then x ∈ (R ∪ S) ∪ T by definition.

References.