Nuprl Lemma : ni-selector-property

`∀p:ℕ∞ ⟶ 𝔹. (∃x:ℕ∞. p x = ff `⇐⇒` p ni-selector(p) = ff)`

Proof

Definitions occuring in Statement :  ni-selector: `ni-selector(p)` nat-inf: `ℕ∞` bfalse: `ff` bool: `𝔹` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` apply: `f a` function: `x:A ⟶ B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` bfalse: `ff` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` exists: `∃x:A. B[x]` nat: `ℕ` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` top: `Top` guard: `{T}` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` assert: `↑b` ifthenelse: `if b then t else f fi ` true: `True` sq_type: `SQType(T)` bnot: `¬bb` less_than: `a < b` nat-inf: `ℕ∞` ni-selector: `ni-selector(p)` nat2inf: `n∞` rev_uimplies: `rev_uimplies(P;Q)` nat-inf-infinity: `∞`
Lemmas referenced :  nat-inf_wf ni-selector_wf bool_wf equal_wf exists_wf equal-wf-T-base nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf assert_witness nat2inf_wf less_than_transitivity1 less_than_irreflexivity int_seg_wf int_seg_properties int_seg_subtype_nat false_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma decidable__equal_int int_seg_subtype intformeq_wf int_formula_prop_eq_lemma le_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot decidable__lt lelt_wf itermAdd_wf int_term_value_add_lemma nat_wf not_wf assert_wf not_assert_elim btrue_neq_bfalse assert_of_lt_int lt_int_wf all_wf iff_imp_equal_bool bnot_wf b-exists_wf assert-b-exists iff_wf assert_of_bnot true_wf equal-nat-inf-infinity and_wf assert_elim
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation cut applyEquality functionExtensionality hypothesisEquality introduction extract_by_obid hypothesis sqequalHypSubstitution isectElimination thin unionElimination equalityElimination equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination sqequalRule lambdaEquality baseClosed dependent_pairFormation functionEquality setElimination rename intWeakElimination natural_numberEquality independent_isectElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll because_Cache productElimination applyLambdaEquality hypothesis_subsumption dependent_set_memberEquality promote_hyp instantiate cumulativity addEquality hyp_replacement addLevel impliesFunctionality impliesLevelFunctionality existsFunctionality existsLevelFunctionality

Latex:
\mforall{}p:\mBbbN{}\minfty{}  {}\mrightarrow{}  \mBbbB{}.  (\mexists{}x:\mBbbN{}\minfty{}.  p  x  =  ff  \mLeftarrow{}{}\mRightarrow{}  p  ni-selector(p)  =  ff)

Date html generated: 2017_10_01-AM-08_29_29
Last ObjectModification: 2017_07_26-PM-04_24_01

Theory : basic

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