Claim L0:
∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ famunion x0 (λ x3 . prim4 (x1 x3)) ⊆ famunion x0 (λ x3 . prim4 (x2 x3))Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3.
Let x3 of type ι be given.
Apply famunionE with
x0,
λ x4 . prim4 (x1 x4),
x3,
x3 ∈ famunion x0 (λ x4 . prim4 (x2 x4)) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Assume H2:
and (x4 ∈ x0) (x3 ∈ prim4 (x1 x4)).
Apply H2 with
x3 ∈ famunion x0 (λ x5 . prim4 (x2 x5)).
Assume H3: x4 ∈ x0.
Assume H4:
x3 ∈ prim4 (x1 x4).
Apply famunionI with
x0,
λ x5 . prim4 (x2 x5),
x4,
x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
λ x5 x6 . x3 ∈ prim4 x5 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Claim L1:
∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ famunion x0 (λ x3 . prim4 (x1 x3)) = famunion x0 (λ x3 . prim4 (x2 x3))Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H1: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3.
Apply set_ext with
famunion x0 (λ x3 . prim4 (x1 x3)),
famunion x0 (λ x3 . prim4 (x2 x3)) leaving 2 subgoals.
Apply L0 with
x0,
x1,
x2.
The subproof is completed by applying H1.
Apply L0 with
x0,
x2,
x1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Let x4 of type ι → ι → ο be given.
Apply H1 with
x3,
λ x5 x6 . x4 x6 x5.
The subproof is completed by applying H2.
Apply In_rec_i_eq with
λ x0 . λ x1 : ι → ι . famunion x0 (λ x2 . prim4 (x1 x2)).
The subproof is completed by applying L1.