α✝ : Type ?u.149199
r : α✝ → α✝ → Prop
β α : Type u_1
f : β → α
w : (#α) < (#β)
w' : ℵ₀ ≤ (#α)
⊢ ∃ a, ℵ₀ ≤ (#↑(f ⁻¹' {a}))
α✝ : Type ?u.149199
r : α✝ → α✝ → Prop
β α : Type u_1
f : β → α
w : (#α) < (#β)
w' : Infinite α
⊢ ∃ a, Infinite ↑(f ⁻¹' {a})
simp_rw [Cardinal.infinite_iff] at w'⊢
case intro
α✝ : Type ?u.149199
r : α✝ → α✝ → Prop
β α : Type u_1
f : β → α
w : (#α) < (#β)
w' : ℵ₀ ≤ (#α)
a : α
ha : (#α) < (#↑(f ⁻¹' {a}))
⊢ ∃ a, ℵ₀ ≤ (#↑(f ⁻¹' {a}))
α✝ : Type ?u.149199
r : α✝ → α✝ → Prop
β α : Type u_1
f : β → α
w : (#α) < (#β)
w' : ℵ₀ ≤ (#α)
⊢ ∃ a, ℵ₀ ≤ (#↑(f ⁻¹' {a}))
cases' infinite_pigeonhole_card_lt f w w' with a ha
no goals
case intro
α✝ : Type ?u.149199
r : α✝ → α✝ → Prop
β α : Type u_1
f : β → α
w : (#α) < (#β)
w' : ℵ₀ ≤ (#α)
a : α
ha : (#α) < (#↑(f ⁻¹' {a}))
⊢ ∃ a, ℵ₀ ≤ (#↑(f ⁻¹' {a}))
exact ⟨a, w'.trans ha.le⟩