Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: x0 ⊆ x1.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H2:
x3 ∈ prim3 x2.
Apply H1 with
x3,
or (ordinal x3) (∃ x4 . and (x4 ∈ x1) (x3 = Sing x4)) leaving 3 subgoals.
The subproof is completed by applying H2.
Apply orIL with
ordinal x3,
∃ x4 . and (x4 ∈ x1) (x3 = Sing x4).
The subproof is completed by applying H3.
Assume H3:
∃ x4 . and (x4 ∈ x0) (x3 = Sing x4).
Apply H3 with
or (ordinal x3) (∃ x4 . and (x4 ∈ x1) (x3 = Sing x4)).
Let x4 of type ι be given.
Assume H4:
(λ x5 . and (x5 ∈ x0) (x3 = Sing x5)) x4.
Apply H4 with
or (ordinal x3) (∃ x5 . and (x5 ∈ x1) (x3 = Sing x5)).
Assume H5: x4 ∈ x0.
Apply orIR with
ordinal x3,
∃ x5 . and (x5 ∈ x1) (x3 = Sing x5).
Let x5 of type ο be given.
Assume H7:
∀ x6 . and (x6 ∈ x1) (x3 = Sing x6) ⟶ x5.
Apply H7 with
x4.
Apply andI with
x4 ∈ x1,
x3 = Sing x4 leaving 2 subgoals.
Apply H0 with
x4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.