Please help with this Lean proof with explanation!

A World: Complement World Level 5 / 5 : Complement subsets equivalence Previous Leave World E Suppose A and B are sets. Then A C B if and only if B C Ac. In this last level of Complement World, example (A B : Set U) : A S B - Bc S Ac := by Tactics you'll prove a statement of the form P - apply by_cases 0 by_contra Q . The most useful theorem for this 1 apply Iff . intro purpose is Iff . intro . If you have h1 : 2 exact compl_subset_compl_of_subset 3 intro h cases' constructor exact P - Q and h2 : Q - P , then Iff . intro 4 intro x h1 h2 is a proof of P - Q . As we saw in 5 intro hx ext have intro left the last level, that means you can start 6 rw [mem compl iff] at hx 7 rw [mem_compl_iff] obtain push_neg rewrite your proof with apply Iff . intro . Lean 8 intro hA will set P - Q and Q - P as the goals 9 apply hx rfl right use that are needed to complete the proof. 10 exact h in hA 11 intro h Current Goal Definitions Objects: A E For the proof in this level, apply Iff. intro will create the goals A S B - U : Type An Bc S Ac and Bc S Ac - ASB . A B : Set U Goal: no Uo C ACB - Bc CAC The second goal is similar, but a little Theorems trickier. For the proof in this level, apply Iff . intro will create the goals A E B - Bc S Ac and Bc g Logic { c nu c novo Ac - ASB. compl_compl The theorem compl_subset_compl_of_subset h compl_subset_compl_of_subset doesn't prove the goal, but it comes mem_compl_iff compl_inter close. Do you see what assertion it will justify? compl_union