Apply In_ind with
λ x0 . ∀ x1 . or (x0 ∈ V_ x1) (V_ x1 ⊆ V_ x0).
Let x0 of type ι be given.
Assume H0:
∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . or (x1 ∈ V_ x2) (V_ x2 ⊆ V_ x1).
Let x1 of type ι be given.
Apply xm with
x0 ∈ V_ x1,
or (x0 ∈ V_ x1) (V_ x1 ⊆ V_ x0) leaving 2 subgoals.
Apply orIL with
x0 ∈ V_ x1,
V_ x1 ⊆ V_ x0.
The subproof is completed by applying H1.
Assume H1:
nIn x0 (V_ x1).
Apply orIR with
x0 ∈ V_ x1,
V_ x1 ⊆ V_ x0.
Let x2 of type ι be given.
Apply V_E with
x2,
x1,
x2 ∈ V_ x0 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x3 of type ι be given.
Assume H3: x3 ∈ x1.
Claim L5:
∃ x4 . and (x4 ∈ x0) (nIn x4 (V_ x3))Apply dneg with
∃ x4 . and (x4 ∈ x0) (nIn x4 (V_ x3)).
Assume H5:
not (∃ x4 . and (x4 ∈ x0) (nIn x4 (V_ x3))).
Apply H1.
Apply V_I with
x0,
x3,
x1 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x4 of type ι be given.
Assume H6: x4 ∈ x0.
Apply dneg with
x4 ∈ V_ x3.
Assume H7:
nIn x4 (V_ x3).
Apply H5.
Let x5 of type ο be given.
Assume H8:
∀ x6 . and (x6 ∈ x0) (nIn x6 (V_ x3)) ⟶ x5.
Apply H8 with
x4.
Apply andI with
x4 ∈ x0,
nIn x4 (V_ x3) leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
Apply L5 with
x2 ∈ V_ x0.
Let x4 of type ι be given.
Assume H6:
(λ x5 . and (x5 ∈ x0) (nIn x5 (V_ x3))) x4.
Apply H6 with
x2 ∈ V_ x0.
Assume H7: x4 ∈ x0.
Assume H8:
nIn x4 (V_ x3).
Apply H0 with
x4,
x3,
x2 ∈ V_ x0 leaving 3 subgoals.
The subproof is completed by applying H7.
Apply FalseE with
x2 ∈ V_ x0.
Apply H8.
The subproof is completed by applying H9.
Assume H9:
V_ x3 ⊆ V_ x4.
Apply Subq_tra with
x2,
V_ x3,
V_ x4 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H9.
Apply V_I with
x2,
x4,
x0 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying L10.