α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a : α
⊢ Bounded r (s ∩ {b | ¬r b a}) → Bounded r s
α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a : α
⊢ Bounded r (s ∩ {b | ¬r b a}) ↔ Bounded r s
refine' ⟨_, Bounded.mono (Set.inter_subset_left s _)⟩
case intro
α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a b : α
hb : ∀ (b_1 : α), b_1 ∈ s ∩ {b | ¬r b a} → r b_1 b
⊢ Bounded r s
α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a : α
⊢ Bounded r (s ∩ {b | ¬r b a}) → Bounded r s
rintro ⟨b, hb⟩
case intro.intro
α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a b : α
hb : ∀ (b_1 : α), b_1 ∈ s ∩ {b | ¬r b a} → r b_1 b
m : α
hm : ∀ (c : α), r c a ∨ r c b → r c m
⊢ Bounded r s
case intro
α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a b : α
hb : ∀ (b_1 : α), b_1 ∈ s ∩ {b | ¬r b a} → r b_1 b
⊢ Bounded r s
cases' H a b with m hm
no goals
case intro.intro
α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a b : α
hb : ∀ (b_1 : α), b_1 ∈ s ∩ {b | ¬r b a} → r b_1 b
m : α
hm : ∀ (c : α), r c a ∨ r c b → r c m
⊢ Bounded r s
exact ⟨m, fun c hc => hm c (or_iff_not_imp_left.2 fun hca => hb c ⟨hc, hca⟩)⟩