### Nuprl Lemma : not-choice-baire-to-nat-ssq

`¬(∀P:((ℕ ⟶ ℕ) ⟶ ℕ) ⟶ ℙ. (∀t:(ℕ ⟶ ℕ) ⟶ ℕ. (↓P[t]) `⇐⇒` ↓∀t:(ℕ ⟶ ℕ) ⟶ ℕ. P[t]))`

Proof

Definitions occuring in Statement :  nat: `ℕ` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` not: `¬A` squash: `↓T` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  not: `¬A` implies: `P `` Q` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` prop: `ℙ` subtype_rel: `A ⊆r B` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` iff: `P `⇐⇒` Q` guard: `{T}` squash: `↓T` unsquashed-WCP: `unsquashed-WCP` exists: `∃x:A. B[x]` rev_implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` nat: `ℕ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top`
Lemmas referenced :  exists_wf nat_wf all_wf equal_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self strong-continuity2-implies-weak-skolem iff_wf squash_wf quotient-implies-squash unsquashed-weak-continuity-false decidable__le nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf le_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis sqequalHypSubstitution dependent_functionElimination thin sqequalRule lambdaEquality introduction extract_by_obid isectElimination functionEquality because_Cache natural_numberEquality applyEquality functionExtensionality hypothesisEquality independent_isectElimination independent_pairFormation independent_functionElimination instantiate cumulativity universeEquality productElimination imageElimination dependent_pairFormation unionElimination equalityTransitivity equalitySymmetry applyLambdaEquality setElimination rename int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll dependent_set_memberEquality

Latex:
\mneg{}(\mforall{}P:((\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.  (\mforall{}t:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}.  (\mdownarrow{}P[t])  \mLeftarrow{}{}\mRightarrow{}  \mdownarrow{}\mforall{}t:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}.  P[t]))

Date html generated: 2017_04_17-AM-10_02_11
Last ObjectModification: 2017_02_27-PM-05_53_38

Theory : continuity

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