Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Assume H0:
∀ x2 . x2 ∈ x0 ⟶ ordinal (x1 x2).
Apply andI with
TransSet (famunion x0 (λ x2 . x1 x2)),
∀ x2 . x2 ∈ famunion x0 (λ x3 . x1 x3) ⟶ TransSet x2 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H1:
x2 ∈ famunion x0 (λ x3 . x1 x3).
Apply famunionE with
x0,
x1,
x2,
x2 ⊆ famunion x0 x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2:
(λ x4 . and (x4 ∈ x0) (x2 ∈ x1 x4)) x3.
Apply H2 with
x2 ⊆ famunion x0 x1.
Assume H3: x3 ∈ x0.
Assume H4: x2 ∈ x1 x3.
Apply H0 with
x3.
The subproof is completed by applying H3.
Apply L5 with
x2 ⊆ famunion x0 (λ x4 . x1 x4).
Assume H7:
∀ x4 . x4 ∈ x1 x3 ⟶ TransSet x4.
Let x4 of type ι be given.
Assume H8: x4 ∈ x2.
Claim L9: x4 ∈ x1 x3
Apply H6 with
x2,
x4 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H8.
Apply famunionI with
x0,
x1,
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying L9.
Let x2 of type ι be given.
Assume H1:
x2 ∈ famunion x0 (λ x3 . x1 x3).
Apply famunionE with
x0,
x1,
x2,
TransSet x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2:
(λ x4 . and (x4 ∈ x0) (x2 ∈ x1 x4)) x3.
Apply H2 with
TransSet x2.
Assume H3: x3 ∈ x0.
Assume H4: x2 ∈ x1 x3.
Apply H0 with
x3.
The subproof is completed by applying H3.
Apply ordinal_Hered with
x1 x3,
x2 leaving 2 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying H4.
Apply L6 with
TransSet x2.
Assume H8:
∀ x4 . x4 ∈ x2 ⟶ TransSet x4.
The subproof is completed by applying H7.