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 → ι be given.
Let x4 of type ι → ι be given.
Apply unknownprop_db24d9aa1dc52b3c0eaf7cf69655226164a8ab5afc5d72e14a32016133f537ca with
x0,
x1,
x3,
bij x0 x2 (λ x5 . x4 (x3 x5)) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H3:
∀ x5 . In x5 x1 ⟶ ∃ x6 . and (In x6 x0) (x3 x6 = x5).
Apply unknownprop_db24d9aa1dc52b3c0eaf7cf69655226164a8ab5afc5d72e14a32016133f537ca with
x1,
x2,
x4,
bij x0 x2 (λ x5 . x4 (x3 x5)) leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H5:
∀ x5 . In x5 x2 ⟶ ∃ x6 . and (In x6 x1) (x4 x6 = x5).
Apply unknownprop_aa42ade5598d8612d2029318c4ed81646c550ecc6cdd9ab953ce4bf73f3dd562 with
x0,
x2,
λ x5 . x4 (x3 x5) leaving 2 subgoals.
Apply unknownprop_a48f98e8977f0c6ed45175c0ee32c0078f43cea0a5c01b41606702afcab1761e with
x0,
x1,
x2,
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Let x5 of type ι be given.
Apply H5 with
x5,
∃ x6 . and (In x6 x0) ((λ x7 . x4 (x3 x7)) x6 = x5) leaving 2 subgoals.
The subproof is completed by applying H6.
Let x6 of type ι be given.
Assume H7:
(λ x7 . and (In x7 x1) (x4 x7 = x5)) x6.
Apply andE with
In x6 x1,
x4 x6 = x5,
∃ x7 . and (In x7 x0) ((λ x8 . x4 (x3 x8)) x7 = x5) leaving 2 subgoals.
The subproof is completed by applying H7.
Assume H9: x4 x6 = x5.
Apply H3 with
x6,
∃ x7 . and (In x7 x0) ((λ x8 . x4 (x3 x8)) x7 = x5) leaving 2 subgoals.
The subproof is completed by applying H8.
Let x7 of type ι be given.
Assume H10:
(λ x8 . and (In x8 x0) (x3 x8 = x6)) x7.
Apply andE with
In x7 x0,
x3 x7 = x6,
∃ x8 . and (In x8 x0) ((λ x9 . x4 (x3 x9)) x8 = x5) leaving 2 subgoals.
The subproof is completed by applying H10.
Assume H12: x3 x7 = x6.
Let x8 of type ο be given.
Assume H13:
∀ x9 . and (In x9 x0) (x4 (x3 x9) = x5) ⟶ x8.
Apply H13 with
x7.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
In x7 x0,
x4 (x3 x7) = x5 leaving 2 subgoals.
The subproof is completed by applying H11.
Apply H12 with
λ x9 x10 . x4 x10 = x5.
The subproof is completed by applying H9.