Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: x1 ∈ x0.
Let x2 of type ι → ι be given.
Assume H1: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0.
Apply nat_ind with
λ x3 . 1319b.. x3 x2 x1 ∈ x0 leaving 2 subgoals.
Apply unknownprop_039fc83525f9619f7cfecb750766b6bca3d944a312bec3cbff47462eeab06c10 with
x2,
x1,
λ x3 x4 . x4 ∈ x0.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Apply unknownprop_002dbea24d6e2c65c6cefd906b209766da62711ebb920e89995e5f3cbbd95f66 with
x3,
x2,
x1,
λ x4 x5 . x5 ∈ x0 leaving 2 subgoals.
Apply nat_p_omega with
x3.
The subproof is completed by applying H2.
Apply H1 with
1319b.. x3 x2 x1.
The subproof is completed by applying H3.