`(∀P:(ℕ ⟶ ℕ) ⟶ ℙ. ∀f:ℕ ⟶ ℕ.  ((P f) `` ⇃(∃k:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕk ⟶ ℕ)) `` (P g)))))`
` (¬(∀A:ℕ ⟶ ℙ. ((∀m:ℕ. (¬¬(A m))) `` (¬¬(∀m:ℕ. (A m))))))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` true: `True` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  implies: `P `` Q` not: `¬A` false: `False` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` is-absolutely-free: `is-absolutely-free{i:l}(f)`
Lemmas referenced :  all_wf nat_wf not_wf quotient_wf exists_wf equal_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self true_wf equiv_rel_true ge_wf zero-seq_wf Kripke2a increasing-zero-seq Kripke2b init0-zero-seq
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin because_Cache hypothesis sqequalHypSubstitution independent_functionElimination voidElimination instantiate introduction extract_by_obid isectElimination functionEquality cumulativity universeEquality sqequalRule lambdaEquality applyEquality functionExtensionality hypothesisEquality natural_numberEquality setElimination rename independent_isectElimination independent_pairFormation dependent_functionElimination intEquality

Latex:
(\mforall{}P:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.    ((P  f)  {}\mRightarrow{}  \00D9(\mexists{}k:\mBbbN{}.  \mforall{}g:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  ((f  =  g)  {}\mRightarrow{}  (P  g)))))
{}\mRightarrow{}  (\mneg{}(\mforall{}A:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}.  ((\mforall{}m:\mBbbN{}.  (\mneg{}\mneg{}(A  m)))  {}\mRightarrow{}  (\mneg{}\mneg{}(\mforall{}m:\mBbbN{}.  (A  m))))))

Date html generated: 2017_09_29-PM-06_09_52
Last ObjectModification: 2017_07_26-PM-03_12_43

Theory : continuity

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