File: Mathlib/Order/SuccPred/CompleteLinearOrder.lean

full_name: csSup_mem_of_not_isSuccLimit
code: lemma csSup_mem_of_not_isSuccLimit
    (hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬ IsSuccLimit (sSup s)) :
    sSup s ∈ s 
goal: sSup s ∈ s

full_name: csInf_mem_of_not_isPredLimit
code: lemma csInf_mem_of_not_isPredLimit
    (hne : s.Nonempty) (hbdd : BddBelow s) (hlim : ¬ IsPredLimit (sInf s)) :
    sInf s ∈ s 
goal: sInf s ∈ s

full_name: exists_eq_ciSup_of_not_isSuccLimit
code: lemma exists_eq_ciSup_of_not_isSuccLimit
    (hf : BddAbove (Set.range f)) (hf' : ¬ IsSuccLimit (⨆ i, f i)) :
    ∃ i, f i = ⨆ i, f i 
goal: ∃ i, f i = ⨆ i, f i

full_name: exists_eq_ciInf_of_not_isPredLimit
code: lemma exists_eq_ciInf_of_not_isPredLimit
    (hf : BddBelow (Set.range f)) (hf' : ¬ IsPredLimit (⨅ i, f i)) :
    ∃ i, f i = ⨅ i, f i 
goal: ∃ i, f i = ⨅ i, f i

full_name: IsLUB.mem_of_nonempty_of_not_isSuccLimit
code: lemma IsLUB.mem_of_nonempty_of_not_isSuccLimit
    (hs : IsLUB s x) (hne : s.Nonempty) (hx : ¬ IsSuccLimit x) : x ∈ s 
goal: x ∈ s

full_name: IsGLB.mem_of_nonempty_of_not_isPredLimit
code: lemma IsGLB.mem_of_nonempty_of_not_isPredLimit
    (hs : IsGLB s x) (hne : s.Nonempty) (hx : ¬ IsPredLimit x) : x ∈ s 
goal: x ∈ s

full_name: IsLUB.exists_of_nonempty_of_not_isSuccLimit
code: lemma IsLUB.exists_of_nonempty_of_not_isSuccLimit
    (hf : IsLUB (Set.range f) x) (hx : ¬ IsSuccLimit x) :
    ∃ i, f i = x 
goal: ∃ i, f i = x

full_name: IsGLB.exists_of_nonempty_of_not_isPredLimit
code: lemma IsGLB.exists_of_nonempty_of_not_isPredLimit
    (hf : IsGLB (Set.range f) x) (hx : ¬ IsPredLimit x) :
    ∃ i, f i = x 
goal: ∃ i, f i = x

full_name: csSup_mem_of_not_isSuccLimit'
code: /-- See `csSup_mem_of_not_isSuccLimit` for the `ConditionallyCompleteLinearOrder` version. -/
lemma csSup_mem_of_not_isSuccLimit'
    (hbdd : BddAbove s) (hlim : ¬ IsSuccLimit (sSup s)) :
    sSup s ∈ s 
goal: sSup s ∈ s

full_name: exists_eq_ciSup_of_not_isSuccLimit'
code: /-- See `exists_eq_ciSup_of_not_isSuccLimit` for the
`ConditionallyCompleteLinearOrder` version. -/
lemma exists_eq_ciSup_of_not_isSuccLimit'
    (hf : BddAbove (Set.range f)) (hf' : ¬ IsSuccLimit (⨆ i, f i)) :
    ∃ i, f i = ⨆ i, f i 
goal: ∃ i, f i = ⨆ i, f i

full_name: IsLUB.mem_of_not_isSuccLimit
code: lemma IsLUB.mem_of_not_isSuccLimit (hs : IsLUB s x) (hx : ¬ IsSuccLimit x) :
    x ∈ s 
goal: x ∈ s

full_name: IsLUB.exists_of_not_isSuccLimit
code: lemma IsLUB.exists_of_not_isSuccLimit (hf : IsLUB (Set.range f) x) (hx : ¬ IsSuccLimit x) :
    ∃ i, f i = x 
goal: ∃ i, f i = x

full_name: sSup_mem_of_not_isSuccLimit
code: lemma sSup_mem_of_not_isSuccLimit (hlim : ¬ IsSuccLimit (sSup s)) :
    sSup s ∈ s 
goal: sSup s ∈ s

full_name: sInf_mem_of_not_isPredLimit
code: lemma sInf_mem_of_not_isPredLimit (hlim : ¬ IsPredLimit (sInf s)) :
    sInf s ∈ s 
goal: sInf s ∈ s

full_name: exists_eq_iSup_of_not_isSuccLimit
code: lemma exists_eq_iSup_of_not_isSuccLimit (hf : ¬ IsSuccLimit (⨆ i, f i)) :
    ∃ i, f i = ⨆ i, f i 
goal: ∃ i, f i = ⨆ i, f i

full_name: exists_eq_iInf_of_not_isPredLimit
code: lemma exists_eq_iInf_of_not_isPredLimit (hf : ¬ IsPredLimit (⨅ i, f i)) :
    ∃ i, f i = ⨅ i, f i 
goal: ∃ i, f i = ⨅ i, f i

full_name: IsGLB.mem_of_not_isPredLimit
code: lemma IsGLB.mem_of_not_isPredLimit (hs : IsGLB s x) (hx : ¬ IsPredLimit x) :
    x ∈ s 
goal: x ∈ s

full_name: IsGLB.exists_of_not_isPredLimit
code: lemma IsGLB.exists_of_not_isPredLimit (hf : IsGLB (Set.range f) x) (hx : ¬ IsPredLimit x) :
    ∃ i, f i = x 
goal: ∃ i, f i = x