### Nuprl Lemma : filter_is_nil_implies2

`∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)}  ⟶ 𝔹].  (∀x∈L.¬↑P[x]) supposing filter(P;L) = [] ∈ ({x:T| (x ∈ L)}  List)`

Proof

Definitions occuring in Statement :  l_all: `(∀x∈L.P[x])` l_member: `(x ∈ l)` filter: `filter(P;l)` nil: `[]` list: `T List` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` not: `¬A` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` l_all: `(∀x∈L.P[x])` not: `¬A` implies: `P `` Q` false: `False` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` less_than: `a < b` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cand: `A c∧ B` subtype_rel: `A ⊆r B`
Lemmas referenced :  equal-wf-T-base list_wf l_member_wf filter_wf2 set_wf bool_wf assert_wf select_wf list-subtype int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf length_wf member_filter select_member sqequal-nil null_nil_lemma btrue_wf member-implies-null-eq-bfalse and_wf equal_wf null_wf btrue_neq_bfalse nil_member
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setEquality cumulativity hypothesisEquality hypothesis functionExtensionality applyEquality baseClosed functionEquality because_Cache sqequalRule lambdaEquality lambdaFormation setElimination rename universeEquality equalityTransitivity equalitySymmetry independent_isectElimination productElimination dependent_functionElimination unionElimination natural_numberEquality dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination dependent_set_memberEquality applyLambdaEquality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}].    (\mforall{}x\mmember{}L.\mneg{}\muparrow{}P[x])  supposing  filter(P;L)  =  []

Date html generated: 2017_04_14-AM-09_26_20
Last ObjectModification: 2017_02_27-PM-04_00_25

Theory : list_1

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