Theorem: Ordinal.mem_closure_tfae
File: Mathlib/SetTheory/Ordinal/Topology.lean

tactic: tfae_have 1 → 2
state_before: s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
⊢ TFAE
    [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a,
      ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a,
      ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a,
      ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a]

tactic: tfae_have 2 → 3
state_before: s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
⊢ TFAE
    [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a,
      ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a,
      ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a,
      ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a]

tactic: tfae_have 3 → 4
state_before: s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a
⊢ TFAE
    [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a,
      ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a,
      ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a,
      ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a]

tactic: tfae_have 4 → 5
state_before: s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a
tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a
⊢ TFAE
    [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a,
      ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a,
      ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a,
      ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a]

tactic: tfae_have 5 → 6
state_before: s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a
tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a
tfae_4_to_5 :
  (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) →
    ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a
⊢ TFAE
    [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a,
      ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a,
      ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a,
      ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a]

tactic: tfae_have 6 → 1
state_before: s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a
tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a
tfae_4_to_5 :
  (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) →
    ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a
tfae_5_to_6 :
  (∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a) →
    ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a
⊢ TFAE
    [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a,
      ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a,
      ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a,
      ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a]

tactic: tfae_finish
state_before: s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a
tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a
tfae_4_to_5 :
  (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) →
    ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a
tfae_5_to_6 :
  (∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a) →
    ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a
tfae_6_to_1 : (∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a) → a ∈ closure s
⊢ TFAE
    [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a,
      ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a,
      ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a,
      ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a]

tactic: simp only [mem_closure_iff_nhdsWithin_neBot, inter_comm s, nhdsWithin_inter', nhds_left_eq_nhds]
state_before: case tfae_1_to_2
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
⊢ a ∈ closure s → a ∈ closure (s ∩ Iic a)

tactic: exact id
state_before: case tfae_1_to_2
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
⊢ Filter.NeBot (𝓝[s] a) → Filter.NeBot (𝓝 a ⊓ Filter.principal s)

tactic: intro h
state_before: case tfae_2_to_3
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
⊢ a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a

tactic: cases' (s ∩ Iic a).eq_empty_or_nonempty with he hne
state_before: case tfae_2_to_3
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
h : a ∈ closure (s ∩ Iic a)
⊢ Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a

tactic: simp [he] at h
state_before: case tfae_2_to_3.inl
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
h : a ∈ closure (s ∩ Iic a)
he : s ∩ Iic a = ∅
⊢ Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a

tactic: refine ⟨hne, (isLUB_of_mem_closure ?_ h).csSup_eq hne⟩
state_before: case tfae_2_to_3.inr
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
h : a ∈ closure (s ∩ Iic a)
hne : Set.Nonempty (s ∩ Iic a)
⊢ Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a

tactic: exact fun x hx => hx.2
state_before: case tfae_2_to_3.inr
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
h : a ∈ closure (s ∩ Iic a)
hne : Set.Nonempty (s ∩ Iic a)
⊢ a ∈ upperBounds (s ∩ Iic a)

tactic: exact fun h => ⟨_, inter_subset_left _ _, h.1, bddAbove_Iic.mono (inter_subset_right _ _), h.2⟩
state_before: case tfae_3_to_4
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a
⊢ Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a

tactic: rintro ⟨t, hts, hne, hbdd, rfl⟩
state_before: case tfae_4_to_5
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a
tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a
⊢ (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) →
    ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a

tactic: have hlub : IsLUB t (sSup t) := isLUB_csSup hne hbdd
state_before: case tfae_4_to_5.intro.intro.intro.intro
s✝ : Set Ordinal
a : Ordinal
s t : Set Ordinal
hts : t ⊆ s
hne : Set.Nonempty t
hbdd : BddAbove t
tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t))
tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t →
    ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t
⊢ ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = sSup t

tactic: let ⟨y, hyt⟩ := hne
state_before: case tfae_4_to_5.intro.intro.intro.intro
s✝ : Set Ordinal
a : Ordinal
s t : Set Ordinal
hts : t ⊆ s
hne : Set.Nonempty t
hbdd : BddAbove t
tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t))
tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t →
    ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t
hlub : IsLUB t (sSup t)
⊢ ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = sSup t

tactic: refine ⟨succ (sSup t), succ_ne_zero _, fun x _ => if x ∈ t then x else y, fun x _ => ?_, ?_⟩
state_before: case tfae_4_to_5.intro.intro.intro.intro
s✝ : Set Ordinal
a : Ordinal
s t : Set Ordinal
hts : t ⊆ s
hne : Set.Nonempty t
hbdd : BddAbove t
tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t))
tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t →
    ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t
hlub : IsLUB t (sSup t)
y : Ordinal
hyt : y ∈ t
⊢ ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = sSup t

tactic: simp only
state_before: case tfae_4_to_5.intro.intro.intro.intro.refine_1
s✝ : Set Ordinal
a : Ordinal
s t : Set Ordinal
hts : t ⊆ s
hne : Set.Nonempty t
hbdd : BddAbove t
tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t))
tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t →
    ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t
hlub : IsLUB t (sSup t)
y : Ordinal
hyt : y ∈ t
x : Ordinal
x✝ : x < succ (sSup t)
⊢ (fun x x_1 => if x ∈ t then x else y) x x✝ ∈ s

tactic: split_ifs with h <;> exact hts ‹_›
state_before: case tfae_4_to_5.intro.intro.intro.intro.refine_1
s✝ : Set Ordinal
a : Ordinal
s t : Set Ordinal
hts : t ⊆ s
hne : Set.Nonempty t
hbdd : BddAbove t
tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t))
tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t →
    ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t
hlub : IsLUB t (sSup t)
y : Ordinal
hyt : y ∈ t
x : Ordinal
x✝ : x < succ (sSup t)
⊢ (if x ∈ t then x else y) ∈ s

tactic: refine le_antisymm (bsup_le fun x _ => ?_) (csSup_le hne fun x hx => ?_)
state_before: case tfae_4_to_5.intro.intro.intro.intro.refine_2
s✝ : Set Ordinal
a : Ordinal
s t : Set Ordinal
hts : t ⊆ s
hne : Set.Nonempty t
hbdd : BddAbove t
tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t))
tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t →
    ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t
hlub : IsLUB t (sSup t)
y : Ordinal
hyt : y ∈ t
⊢ (bsup (succ (sSup t)) fun x x_1 => if x ∈ t then x else y) = sSup t

tactic: split_ifs <;> exact hlub.1 ‹_›
state_before: case tfae_4_to_5.intro.intro.intro.intro.refine_2.refine_1
s✝ : Set Ordinal
a : Ordinal
s t : Set Ordinal
hts : t ⊆ s
hne : Set.Nonempty t
hbdd : BddAbove t
tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t))
tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t →
    ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t
hlub : IsLUB t (sSup t)
y : Ordinal
hyt : y ∈ t
x : Ordinal
x✝ : x < succ (sSup t)
⊢ (if x ∈ t then x else y) ≤ sSup t

tactic: refine (if_pos hx).symm.trans_le (le_bsup _ _ <| (hlub.1 hx).trans_lt (lt_succ _))
state_before: case tfae_4_to_5.intro.intro.intro.intro.refine_2.refine_2
s✝ : Set Ordinal
a : Ordinal
s t : Set Ordinal
hts : t ⊆ s
hne : Set.Nonempty t
hbdd : BddAbove t
tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t))
tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t →
    ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t
hlub : IsLUB t (sSup t)
y : Ordinal
hyt : y ∈ t
x : Ordinal
hx : x ∈ t
⊢ x ≤ bsup (succ (sSup t)) fun x x_1 => if x ∈ t then x else y

tactic: rintro ⟨o, h₀, f, hfs, rfl⟩
state_before: case tfae_5_to_6
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a
tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a
tfae_4_to_5 :
  (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) →
    ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a
⊢ (∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a) →
    ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a

tactic: exact ⟨_, out_nonempty_iff_ne_zero.2 h₀, familyOfBFamily o f, fun _ => hfs _ _, rfl⟩
state_before: case tfae_5_to_6.intro.intro.intro.intro
s✝ : Set Ordinal
a : Ordinal
s : Set Ordinal
o : Ordinal
h₀ : o ≠ 0
f : (x : Ordinal) → x < o → Ordinal
hfs : ∀ (x : Ordinal) (hx : x < o), f x hx ∈ s
tfae_1_to_2 : bsup o f ∈ closure s → bsup o f ∈ closure (s ∩ Iic (bsup o f))
tfae_2_to_3 :
  bsup o f ∈ closure (s ∩ Iic (bsup o f)) → Set.Nonempty (s ∩ Iic (bsup o f)) ∧ sSup (s ∩ Iic (bsup o f)) = bsup o f
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (bsup o f)) ∧ sSup (s ∩ Iic (bsup o f)) = bsup o f →
    ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = bsup o f
tfae_4_to_5 :
  (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = bsup o f) →
    ∃ o_1, o_1 ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal) (hx : x < o_1), f_1 x hx ∈ s) ∧ bsup o_1 f_1 = bsup o f
⊢ ∃ ι, Nonempty ι ∧ ∃ f_1, (∀ (i : ι), f_1 i ∈ s) ∧ sup f_1 = bsup o f

tactic: rintro ⟨ι, hne, f, hfs, rfl⟩
state_before: case tfae_6_to_1
s✝ : Set Ordinal
a✝ a : Ordinal
s : Set Ordinal
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a
tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a
tfae_4_to_5 :
  (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) →
    ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a
tfae_5_to_6 :
  (∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a) →
    ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a
⊢ (∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a) → a ∈ closure s

tactic: rw [sup, iSup]
state_before: case tfae_6_to_1.intro.intro.intro.intro
s✝ : Set Ordinal
a : Ordinal
s : Set Ordinal
ι : Type u
hne : Nonempty ι
f : ι → Ordinal
hfs : ∀ (i : ι), f i ∈ s
tfae_1_to_2 : sup f ∈ closure s → sup f ∈ closure (s ∩ Iic (sup f))
tfae_2_to_3 : sup f ∈ closure (s ∩ Iic (sup f)) → Set.Nonempty (s ∩ Iic (sup f)) ∧ sSup (s ∩ Iic (sup f)) = sup f
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sup f)) ∧ sSup (s ∩ Iic (sup f)) = sup f →
    ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = sup f
tfae_4_to_5 :
  (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = sup f) →
    ∃ o, o ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal) (hx : x < o), f_1 x hx ∈ s) ∧ bsup o f_1 = sup f
tfae_5_to_6 :
  (∃ o, o ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal) (hx : x < o), f_1 x hx ∈ s) ∧ bsup o f_1 = sup f) →
    ∃ ι_1, Nonempty ι_1 ∧ ∃ f_1, (∀ (i : ι_1), f_1 i ∈ s) ∧ sup f_1 = sup f
⊢ sup f ∈ closure s

tactic: exact closure_mono (range_subset_iff.2 hfs) <| csSup_mem_closure (range_nonempty f)
  (bddAbove_range.{u, u} f)
state_before: case tfae_6_to_1.intro.intro.intro.intro
s✝ : Set Ordinal
a : Ordinal
s : Set Ordinal
ι : Type u
hne : Nonempty ι
f : ι → Ordinal
hfs : ∀ (i : ι), f i ∈ s
tfae_1_to_2 : sup f ∈ closure s → sup f ∈ closure (s ∩ Iic (sup f))
tfae_2_to_3 : sup f ∈ closure (s ∩ Iic (sup f)) → Set.Nonempty (s ∩ Iic (sup f)) ∧ sSup (s ∩ Iic (sup f)) = sup f
tfae_3_to_4 :
  Set.Nonempty (s ∩ Iic (sup f)) ∧ sSup (s ∩ Iic (sup f)) = sup f →
    ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = sup f
tfae_4_to_5 :
  (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = sup f) →
    ∃ o, o ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal) (hx : x < o), f_1 x hx ∈ s) ∧ bsup o f_1 = sup f
tfae_5_to_6 :
  (∃ o, o ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal) (hx : x < o), f_1 x hx ∈ s) ∧ bsup o f_1 = sup f) →
    ∃ ι_1, Nonempty ι_1 ∧ ∃ f_1, (∀ (i : ι_1), f_1 i ∈ s) ∧ sup f_1 = sup f
⊢ sSup (Set.range f) ∈ closure s