chore(set_theory/*): use is_empty α instead of ¬nonempty α (#8276)

Split from #7826

Author: eric-wieser

src/set_theory/cardinal.lean

@@ -457,8 +457,8 @@ sup_le.2 $ le_sum _
 theorem sum_le_sup {ι : Type u} (f : ι → cardinal.{u}) : sum f ≤ mk ι * sup.{u u} f :=
 by rw ← sum_const; exact sum_le_sum _ _ (le_sup _)
 
-theorem sup_eq_zero {ι} {f : ι → cardinal} (h : ι → false) : sup f = 0 :=
-by { rw [← nonpos_iff_eq_zero, sup_le], intro x, exfalso, exact h x }
+theorem sup_eq_zero {ι} {f : ι → cardinal} [is_empty ι] : sup f = 0 :=
+by { rw [← nonpos_iff_eq_zero, sup_le], exact is_empty_elim }
 
 /-- The indexed product of cardinals is the cardinality of the Pi type
   (dependent product). -/
@@ -1241,7 +1241,10 @@ begin
 end
 
 lemma powerlt_zero {a : cardinal} : a ^< 0 = 0 :=
-by { apply sup_eq_zero, rintro ⟨x, hx⟩, rw [←not_le] at hx, apply hx, apply zero_le }
+begin
+  convert sup_eq_zero,
+  exact subtype.is_empty_of_false (λ x, (zero_le _).not_lt),
+end
 
 end cardinal
 

src/set_theory/cofinality.lean

@@ -195,9 +195,9 @@ le_antisymm (by simpa using cof_le_card 0) (cardinal.zero_le _)
 
 @[simp] theorem cof_eq_zero {o} : cof o = 0 ↔ o = 0 :=
 ⟨induction_on o $ λ α r _ z, by exactI
-  let ⟨S, hl, e⟩ := cof_eq r in type_eq_zero_iff_empty.2 $
-  λ ⟨a⟩, let ⟨b, h, _⟩ := hl a in
-  ne_zero_iff_nonempty.2 (by exact ⟨⟨_, h⟩⟩) (e.trans z),
+  let ⟨S, hl, e⟩ := cof_eq r in type_eq_zero_iff_is_empty.2 $
+  ⟨λ a, let ⟨b, h, _⟩ := hl a in
+    (eq_zero_iff_is_empty.1 (e.trans z)).elim' ⟨_, h⟩⟩,
 λ e, by simp [e]⟩
 
 @[simp] theorem cof_succ (o) : cof (succ o) = 1 :=

src/set_theory/ordinal_arithmetic.lean

@@ -155,12 +155,15 @@ by simp only [le_antisymm_iff, add_le_add_iff_right]
   exact ⟨f punit.star⟩
 end, λ e, by simp only [e, card_zero]⟩
 
+@[simp] theorem type_eq_zero_of_empty [is_well_order α r] [is_empty α] : type r = 0 :=
+card_eq_zero.symm.mpr eq_zero_of_is_empty
+
+@[simp] theorem type_eq_zero_iff_is_empty [is_well_order α r] : type r = 0 ↔ is_empty α :=
+(@card_eq_zero (type r)).symm.trans eq_zero_iff_is_empty
+
 theorem type_ne_zero_iff_nonempty [is_well_order α r] : type r ≠ 0 ↔ nonempty α :=
 (not_congr (@card_eq_zero (type r))).symm.trans ne_zero_iff_nonempty
 
-@[simp] theorem type_eq_zero_iff_empty [is_well_order α r] : type r = 0 ↔ ¬ nonempty α :=
-(not_iff_comm.1 type_ne_zero_iff_nonempty).symm
-
 protected lemma one_ne_zero : (1 : ordinal) ≠ 0 :=
 type_ne_zero_iff_nonempty.2 ⟨punit.star⟩
 
@@ -547,12 +550,10 @@ quotient.induction_on₂ a b $ λ ⟨α, r, _⟩ ⟨β, s, _⟩,
 mul_comm (mk β) (mk α)
 
 @[simp] theorem mul_zero (a : ordinal) : a * 0 = 0 :=
-induction_on a $ λ α _ _, by exactI
-type_eq_zero_iff_empty.2 (λ ⟨⟨e, _⟩⟩, e.elim)
+induction_on a $ λ α _ _, by exactI type_eq_zero_of_empty
 
 @[simp] theorem zero_mul (a : ordinal) : 0 * a = 0 :=
-induction_on a $ λ α _ _, by exactI
-type_eq_zero_iff_empty.2 (λ ⟨⟨_, e⟩⟩, e.elim)
+induction_on a $ λ α _ _, by exactI type_eq_zero_of_empty
 
 theorem mul_add (a b c : ordinal) : a * (b + c) = a * b + a * c :=
 quotient.induction_on₃ a b c $ λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩,