Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Assume H1: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x2 ⟶ x1 x2 ⟶ x1 x3.
Apply andI with
TransSet {x2 ∈ x0|x1 x2},
∀ x2 . x2 ∈ {x3 ∈ x0|x1 x3} ⟶ TransSet x2 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H2: x2 ∈ {x3 ∈ x0|x1 x3}.
Let x3 of type ι be given.
Assume H3: x3 ∈ x2.
Apply SepE with
x0,
x1,
x2,
x3 ∈ {x4 ∈ x0|x1 x4} leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H4: x2 ∈ x0.
Assume H5: x1 x2.
Apply SepI with
x0,
x1,
x3 leaving 2 subgoals.
Apply H0 with
x3 ∈ x0.
Assume H7:
∀ x4 . x4 ∈ x0 ⟶ TransSet x4.
Apply H6 with
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
Apply H1 with
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
Let x2 of type ι be given.
Assume H2: x2 ∈ {x3 ∈ x0|x1 x3}.
Claim L3: x2 ∈ x0
Apply SepE1 with
x0,
x1,
x2.
The subproof is completed by applying H2.
Apply ordinal_Hered with
x0,
x2,
TransSet x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L3.
Assume H5:
∀ x3 . x3 ∈ x2 ⟶ TransSet x3.
The subproof is completed by applying H4.