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Update Basic.lean
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vihdzp committed Dec 4, 2024
1 parent d44c9cb commit 318316d
Showing 1 changed file with 5 additions and 5 deletions.
10 changes: 5 additions & 5 deletions Mathlib/SetTheory/Cardinal/Basic.lean
Original file line number Diff line number Diff line change
Expand Up @@ -845,7 +845,7 @@ end deprecated
is a strong limit by this definition. -/
structure IsStrongLimit (c : Cardinal) : Prop where
ne_zero : c ≠ 0
two_power_lt : ∀ x < c, 2 ^ x < c
two_power_lt {x} : x < c 2 ^ x < c

protected theorem IsStrongLimit.isSuccLimit {c} (H : IsStrongLimit c) : IsSuccLimit c := by
rw [Cardinal.isSuccLimit_iff]
Expand Down Expand Up @@ -1540,10 +1540,10 @@ theorem aleph0_le_of_isSuccLimit {c : Cardinal} (h : IsSuccLimit c) : ℵ₀ ≤
contrapose! h
exact not_isSuccLimit_of_lt_aleph0 h

theorem isStrongLimit_aleph0 : IsStrongLimit ℵ₀ :=
⟨aleph0_ne_zero, fun x hx => by
rcases lt_aleph0.1 hx with ⟨n, rfl⟩
exact mod_cast nat_lt_aleph0 (2 ^ n)⟩
theorem isStrongLimit_aleph0 : IsStrongLimit ℵ₀ := by
refine ⟨aleph0_ne_zero, fun hx ↦ ?_⟩
obtain ⟨n, rfl⟩ := lt_aleph0.1 hx
exact_mod_cast nat_lt_aleph0 _

theorem IsStrongLimit.aleph0_le {c} (H : IsStrongLimit c) : ℵ₀ ≤ c :=
aleph0_le_of_isSuccLimit H.isSuccLimit
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