File: Mathlib/SetTheory/Cardinal/Continuum.lean

full_name: Cardinal.continuum
code: /-- Cardinality of continuum. -/
def continuum : Cardinal.{u} 
goal: Cardinal.{u}

full_name: Cardinal._aux_Mathlib_SetTheory_Cardinal_Continuum___macroRules_Cardinal_term𝔠_1
code: scoped notation "𝔠" => Cardinal.continuum
goal: Lean.Macro

full_name: Cardinal._aux_Mathlib_SetTheory_Cardinal_Continuum___unexpand_Cardinal_continuum_1
code: scoped notation "𝔠" => Cardinal.continuum
goal: Lean.PrettyPrinter.Unexpander

full_name: Cardinal.two_power_aleph0
code: @[simp]
theorem two_power_aleph0 : 2 ^ aleph0.{u} = continuum.{u} 
goal: 2 ^ Cardinal.aleph0 = Cardinal.continuum

full_name: Cardinal.lift_continuum
code: @[simp]
theorem lift_continuum : lift.{v} 𝔠 = 𝔠 
goal: Cardinal.lift.{v, u_1} Cardinal.continuum = Cardinal.continuum

full_name: Cardinal.continuum_le_lift
code: @[simp]
theorem continuum_le_lift {c : Cardinal.{u}} : 𝔠 ≤ lift.{v} c ↔ 𝔠 ≤ c 
goal: Cardinal.continuum ≤ Cardinal.lift.{v, u} c ↔ Cardinal.continuum ≤ c

full_name: Cardinal.lift_le_continuum
code: @[simp]
theorem lift_le_continuum {c : Cardinal.{u}} : lift.{v} c ≤ 𝔠 ↔ c ≤ 𝔠 
goal: Cardinal.lift.{v, u} c ≤ Cardinal.continuum ↔ c ≤ Cardinal.continuum

full_name: Cardinal.continuum_lt_lift
code: @[simp]
theorem continuum_lt_lift {c : Cardinal.{u}} : 𝔠 < lift.{v} c ↔ 𝔠 < c 
goal: Cardinal.continuum < Cardinal.lift.{v, u} c ↔ Cardinal.continuum < c

full_name: Cardinal.lift_lt_continuum
code: @[simp]
theorem lift_lt_continuum {c : Cardinal.{u}} : lift.{v} c < 𝔠 ↔ c < 𝔠 
goal: Cardinal.lift.{v, u} c < Cardinal.continuum ↔ c < Cardinal.continuum

full_name: Cardinal.aleph0_lt_continuum
code: theorem aleph0_lt_continuum : ℵ₀ < 𝔠 
goal: Cardinal.aleph0 < Cardinal.continuum

full_name: Cardinal.aleph0_le_continuum
code: theorem aleph0_le_continuum : ℵ₀ ≤ 𝔠 
goal: Cardinal.aleph0 ≤ Cardinal.continuum

full_name: Cardinal.beth_one
code: @[simp]
theorem beth_one : beth 1 = 𝔠 
goal: Cardinal.beth 1 = Cardinal.continuum

full_name: Cardinal.nat_lt_continuum
code: theorem nat_lt_continuum (n : ℕ) : ↑n < 𝔠 
goal: ↑n < Cardinal.continuum

full_name: Cardinal.mk_set_nat
code: theorem mk_set_nat : #(Set ℕ) = 𝔠 
goal: Cardinal.mk (Set ℕ) = Cardinal.continuum

full_name: Cardinal.continuum_pos
code: theorem continuum_pos : 0 < 𝔠 
goal: 0 < Cardinal.continuum

full_name: Cardinal.continuum_ne_zero
code: theorem continuum_ne_zero : 𝔠 ≠ 0 
goal: Cardinal.continuum ≠ 0

full_name: Cardinal.aleph_one_le_continuum
code: theorem aleph_one_le_continuum : aleph 1 ≤ 𝔠 
goal: Cardinal.aleph 1 ≤ Cardinal.continuum

full_name: Cardinal.continuum_toNat
code: @[simp]
theorem continuum_toNat : toNat continuum = 0 
goal: Cardinal.toNat Cardinal.continuum = 0

full_name: Cardinal.continuum_toPartENat
code: @[simp]
theorem continuum_toPartENat : toPartENat continuum = ⊤ 
goal: Cardinal.toPartENat Cardinal.continuum = ⊤

full_name: Cardinal.aleph0_add_continuum
code: @[simp]
theorem aleph0_add_continuum : ℵ₀ + 𝔠 = 𝔠 
goal: Cardinal.aleph0 + Cardinal.continuum = Cardinal.continuum

full_name: Cardinal.continuum_add_aleph0
code: @[simp]
theorem continuum_add_aleph0 : 𝔠 + ℵ₀ = 𝔠 
goal: Cardinal.continuum + Cardinal.aleph0 = Cardinal.continuum

full_name: Cardinal.continuum_add_self
code: @[simp]
theorem continuum_add_self : 𝔠 + 𝔠 = 𝔠 
goal: Cardinal.continuum + Cardinal.continuum = Cardinal.continuum

full_name: Cardinal.nat_add_continuum
code: @[simp]
theorem nat_add_continuum (n : ℕ) : ↑n + 𝔠 = 𝔠 
goal: ↑n + Cardinal.continuum = Cardinal.continuum

full_name: Cardinal.continuum_add_nat
code: @[simp]
theorem continuum_add_nat (n : ℕ) : 𝔠 + n = 𝔠 
goal: Cardinal.continuum + ↑n = Cardinal.continuum

full_name: Cardinal.ofNat_add_continuum
code: @[simp]
theorem ofNat_add_continuum {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) + 𝔠 = 𝔠 
goal: OfNat.ofNat n + Cardinal.continuum = Cardinal.continuum

full_name: Cardinal.continuum_add_ofNat
code: @[simp]
theorem continuum_add_ofNat {n : ℕ} [Nat.AtLeastTwo n] : 𝔠 + no_index (OfNat.ofNat n) = 𝔠 
goal: Cardinal.continuum + OfNat.ofNat n = Cardinal.continuum

full_name: Cardinal.continuum_mul_self
code: @[simp]
theorem continuum_mul_self : 𝔠 * 𝔠 = 𝔠 
goal: Cardinal.continuum * Cardinal.continuum = Cardinal.continuum

full_name: Cardinal.continuum_mul_aleph0
code: @[simp]
theorem continuum_mul_aleph0 : 𝔠 * ℵ₀ = 𝔠 
goal: Cardinal.continuum * Cardinal.aleph0 = Cardinal.continuum

full_name: Cardinal.aleph0_mul_continuum
code: @[simp]
theorem aleph0_mul_continuum : ℵ₀ * 𝔠 = 𝔠 
goal: Cardinal.aleph0 * Cardinal.continuum = Cardinal.continuum

full_name: Cardinal.nat_mul_continuum
code: @[simp]
theorem nat_mul_continuum {n : ℕ} (hn : n ≠ 0) : ↑n * 𝔠 = 𝔠 
goal: ↑n * Cardinal.continuum = Cardinal.continuum

full_name: Cardinal.continuum_mul_nat
code: @[simp]
theorem continuum_mul_nat {n : ℕ} (hn : n ≠ 0) : 𝔠 * n = 𝔠 
goal: Cardinal.continuum * ↑n = Cardinal.continuum

full_name: Cardinal.ofNat_mul_continuum
code: @[simp]
theorem ofNat_mul_continuum {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) * 𝔠 = 𝔠 
goal: OfNat.ofNat n * Cardinal.continuum = Cardinal.continuum

full_name: Cardinal.continuum_mul_ofNat
code: @[simp]
theorem continuum_mul_ofNat {n : ℕ} [Nat.AtLeastTwo n] : 𝔠 * no_index (OfNat.ofNat n) = 𝔠 
goal: Cardinal.continuum * OfNat.ofNat n = Cardinal.continuum

full_name: Cardinal.aleph0_power_aleph0
code: @[simp]
theorem aleph0_power_aleph0 : aleph0.{u} ^ aleph0.{u} = 𝔠 
goal: Cardinal.aleph0 ^ Cardinal.aleph0 = Cardinal.continuum

full_name: Cardinal.nat_power_aleph0
code: @[simp]
theorem nat_power_aleph0 {n : ℕ} (hn : 2 ≤ n) : (n ^ aleph0.{u} : Cardinal.{u}) = 𝔠 
goal: ↑n ^ Cardinal.aleph0 = Cardinal.continuum

full_name: Cardinal.continuum_power_aleph0
code: @[simp]
theorem continuum_power_aleph0 : continuum.{u} ^ aleph0.{u} = 𝔠 
goal: Cardinal.continuum ^ Cardinal.aleph0 = Cardinal.continuum