Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: x1 ∈ x0.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Let x5 of type ι be given.
Assume H4: x5 ∈ x0.
Assume H5: x1 = x2 ⟶ ∀ x6 : ο . x6.
Assume H6: x1 = x3 ⟶ ∀ x6 : ο . x6.
Assume H7: x2 = x3 ⟶ ∀ x6 : ο . x6.
Assume H8: x1 = x4 ⟶ ∀ x6 : ο . x6.
Assume H9: x2 = x4 ⟶ ∀ x6 : ο . x6.
Assume H10: x3 = x4 ⟶ ∀ x6 : ο . x6.
Assume H11: x1 = x5 ⟶ ∀ x6 : ο . x6.
Assume H12: x2 = x5 ⟶ ∀ x6 : ο . x6.
Assume H13: x3 = x5 ⟶ ∀ x6 : ο . x6.
Assume H14: x4 = x5 ⟶ ∀ x6 : ο . x6.
Apply setminusI with
x0,
Sing x5,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H15:
x1 ∈ Sing x5.
Apply H11.
Apply SingE with
x5,
x1.
The subproof is completed by applying H15.
Apply setminusI with
x0,
Sing x5,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H16:
x2 ∈ Sing x5.
Apply H12.
Apply SingE with
x5,
x2.
The subproof is completed by applying H16.
Apply setminusI with
x0,
Sing x5,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H17:
x3 ∈ Sing x5.
Apply H13.
Apply SingE with
x5,
x3.
The subproof is completed by applying H17.
Apply setminusI with
x0,
Sing x5,
x4 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H18:
x4 ∈ Sing x5.
Apply H14.
Apply SingE with
x5,
x4.
The subproof is completed by applying H18.
Apply setminus_nIn_I2 with
x0,
Sing x5,
x5.
The subproof is completed by applying SingI with x5.
Apply unknownprop_b67f105a533ad0f9d1429aa46873f001e12ec9da326933796dd3f178108aca0d with
setminus x0 (Sing x5),
x1,
x2,
x3,
x4 leaving 10 subgoals.
The subproof is completed by applying L15.
The subproof is completed by applying L16.
The subproof is completed by applying L17.
The subproof is completed by applying L18.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply unknownprop_20fce6fc7f2e036c1229cbf996632439eddb19cfae541105a83e5be9c65bc111 with
x0,
x5,
λ x6 x7 . atleastp u5 x7 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply unknownprop_11c6158bd93dbd27daaa9a84a43404be6ccbf75f900b1e28dfa453e64ea6c96b with
u4,
setminus x0 (Sing x5),
x5 leaving 2 subgoals.
The subproof is completed by applying L19.
The subproof is completed by applying L20.