Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply unknownprop_6e9d790c24657bc527a0f62de036403ca00386b366ddc02915f8c3a4de529eee with
In x1 (Union x0),
∃ x3 . and (In x1 x3) (In x3 x0),
In x1 (Union x0) leaving 3 subgoals.
The subproof is completed by applying unknownprop_0ec9d99c25b180996bbc4b628f77b020efad5688fb0f39be5eb6c4c59a6163b1 with x0, x1.
Assume H3:
∃ x3 . and (In x1 x3) (In x3 x0).
The subproof is completed by applying H2.
Assume H3:
not (∃ x3 . and (In x1 x3) (In x3 x0)).
Apply FalseE with
In x1 (Union x0).
Apply notE with
∃ x3 . and (In x1 x3) (In x3 x0) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ο be given.
Assume H4:
∀ x4 . and (In x1 x4) (In x4 x0) ⟶ x3.
Apply H4 with
x2.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
In x1 x2,
In x2 x0 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.