Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Apply H2 with
In2_lexicographic x0 x1 x4 x5 leaving 2 subgoals.
Assume H4: x1 ∈ x3.
Apply H3 with
In2_lexicographic x0 x1 x4 x5 leaving 2 subgoals.
Assume H5: x3 ∈ x5.
Apply orIL with
x1 ∈ x5,
and (x1 = x5) (x0 ∈ x4).
Apply H1 with
x3,
x1 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H4.
Assume H5:
and (x3 = x5) (x2 ∈ x4).
Apply H5 with
In2_lexicographic x0 x1 x4 x5.
Assume H6: x3 = x5.
Assume H7: x2 ∈ x4.
Apply orIL with
x1 ∈ x5,
and (x1 = x5) (x0 ∈ x4).
Apply H6 with
λ x6 x7 . x1 ∈ x6.
The subproof is completed by applying H4.
Assume H4:
and (x1 = x3) (x0 ∈ x2).
Apply H4 with
In2_lexicographic x0 x1 x4 x5.
Assume H5: x1 = x3.
Assume H6: x0 ∈ x2.
Apply H3 with
In2_lexicographic x0 x1 x4 x5 leaving 2 subgoals.
Assume H7: x3 ∈ x5.
Apply orIL with
x1 ∈ x5,
and (x1 = x5) (x0 ∈ x4).
Apply H5 with
λ x6 x7 . x7 ∈ x5.
The subproof is completed by applying H7.
Assume H7:
and (x3 = x5) (x2 ∈ x4).
Apply H7 with
In2_lexicographic x0 x1 x4 x5.
Assume H8: x3 = x5.
Assume H9: x2 ∈ x4.
Apply orIR with
x1 ∈ x5,
and (x1 = x5) (x0 ∈ x4).
Apply andI with
x1 = x5,
x0 ∈ x4 leaving 2 subgoals.
Apply H8 with
λ x6 x7 . x1 = x6.
The subproof is completed by applying H5.
Apply H0 with
x2,
x0 leaving 2 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H6.