Let x0 of type ι be given.
Let x1 of type ι be given.
Apply H0 with
ordinal (binintersect x0 x1).
Assume H3:
∀ x2 . x2 ∈ x0 ⟶ TransSet x2.
Apply H1 with
ordinal (binintersect x0 x1).
Assume H5:
∀ x2 . x2 ∈ x1 ⟶ TransSet x2.
Apply ordinal_In_Or_Subq with
x0,
x1,
ordinal (binintersect x0 x1) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H6: x0 ∈ x1.
Apply binintersect_Subq_eq_1 with
x0,
x1,
λ x2 x3 . ordinal x3 leaving 2 subgoals.
Apply H4 with
x0.
The subproof is completed by applying H6.
The subproof is completed by applying H0.
Assume H6: x1 ⊆ x0.
Apply binintersect_com with
x0,
x1,
λ x2 x3 . ordinal x3.
Apply binintersect_Subq_eq_1 with
x1,
x0,
λ x2 x3 . ordinal x3 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H1.