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fix merge
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vihdzp committed Feb 22, 2025
1 parent 0216ec2 commit 5405d5c
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Showing 2 changed files with 6 additions and 6 deletions.
6 changes: 3 additions & 3 deletions Mathlib/SetTheory/Cardinal/Basic.lean
Original file line number Diff line number Diff line change
Expand Up @@ -850,11 +850,11 @@ alias isSuccLimit_zero := isSuccPrelimit_zero

end deprecated

/-- A cardinal is a strong limit if it is not zero and it is closed under powersets. Note that `ℵ₀`
is a strong limit by this definition. -/
/-- A cardinal is a strong limit if it is not zero and it is closed under powersets.
Note that `ℵ₀` is a strong limit by this definition. -/
structure IsStrongLimit (c : Cardinal) : Prop where
ne_zero : c ≠ 0
two_power_lt {x} : 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]
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6 changes: 3 additions & 3 deletions Mathlib/SetTheory/Cardinal/Cofinality.lean
Original file line number Diff line number Diff line change
Expand Up @@ -806,7 +806,7 @@ theorem mk_bounded_subset {α : Type*} (h : ∀ x < #α, 2 ^ x < #α) {r : α
constructor
rintro ⟨s, hs⟩
exact (not_unbounded_iff s).2 hs (unbounded_of_isEmpty s)
have h' : IsStrongLimit #α := ⟨ha, @h⟩
have h' : IsStrongLimit #α := ⟨ha, h⟩
have ha := h'.aleph0_le
apply le_antisymm
· have : { s : Set α | Bounded r s } = ⋃ i, 𝒫{ j | r j i } := setOf_exists _
Expand All @@ -831,7 +831,7 @@ theorem mk_subset_mk_lt_cof {α : Type*} (h : ∀ x < #α, 2 ^ x < #α) :
#{ s : Set α // #s < cof (#α).ord } = #α := by
rcases eq_or_ne #α 0 with (ha | ha)
· simp [ha]
have h' : IsStrongLimit #α := ⟨ha, @h⟩
have h' : IsStrongLimit #α := ⟨ha, h⟩
rcases ord_eq α with ⟨r, wo, hr⟩
haveI := wo
apply le_antisymm
Expand Down Expand Up @@ -1146,7 +1146,7 @@ def IsInaccessible (c : Cardinal) :=

theorem IsInaccessible.mk {c} (h₁ : ℵ₀ < c) (h₂ : c ≤ c.ord.cof) (h₃ : ∀ x < c, 2 ^ x < c) :
IsInaccessible c :=
⟨h₁, ⟨h₁.le, h₂⟩, (aleph0_pos.trans h₁).ne', @h₃⟩
⟨h₁, ⟨h₁.le, h₂⟩, (aleph0_pos.trans h₁).ne', h₃⟩

-- Lean's foundations prove the existence of ℵ₀ many inaccessible cardinals
theorem univ_inaccessible : IsInaccessible univ.{u, v} :=
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