Let x0 of type ι be given.
Let x1 of type ι be given.
Apply dneg with
∃ x2 . and (x2 ∈ x0) (x1 ∈ x2).
Assume H3:
not (∃ x2 . and (x2 ∈ x0) (x1 ∈ x2)).
Apply H1.
Apply Subq_finite with
ordsucc x1,
x0 leaving 2 subgoals.
Apply unknownprop_a27dbe9bdb3250ff525cd2b00221a19841e7e622da131f38fcc1540ba15eb9d8 with
ordsucc x1.
Apply nat_ordsucc with
x1.
Apply omega_nat_p with
x1.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Assume H4: x2 ∈ x0.
Apply ordinal_trichotomy_or_impred with
x1,
x2,
x2 ∈ ordsucc x1 leaving 5 subgoals.
Apply nat_p_ordinal with
x1.
Apply omega_nat_p with
x1.
The subproof is completed by applying H2.
Apply nat_p_ordinal with
x2.
Apply omega_nat_p with
x2.
Apply H0 with
x2.
The subproof is completed by applying H4.
Assume H5: x1 ∈ x2.
Apply FalseE with
x2 ∈ ordsucc x1.
Apply H3.
Let x3 of type ο be given.
Assume H6:
∀ x4 . and (x4 ∈ x0) (x1 ∈ x4) ⟶ x3.
Apply H6 with
x2.
Apply andI with
x2 ∈ x0,
x1 ∈ x2 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Assume H5: x1 = x2.
Apply H5 with
λ x3 x4 . x3 ∈ ordsucc x1.
The subproof is completed by applying ordsuccI2 with x1.
The subproof is completed by applying ordsuccI1 with x1, x2.