Nuprl Lemma : list-set-type-property

[T:Type]. ∀[P:T ⟶ ℙ].  ((∀x:T. SqStable(P[x]))  (∀L:{x:T| P[x]}  List. (∀x∈L.P[x])))


Proof




Definitions occuring in Statement :  l_all: (∀x∈L.P[x]) list: List sq_stable: SqStable(P) uall: [x:A]. B[x] prop: so_apply: x[s] all: x:A. B[x] implies:  Q set: {x:A| B[x]}  function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] implies:  Q all: x:A. B[x] l_all: (∀x∈L.P[x]) member: t ∈ T so_apply: x[s] subtype_rel: A ⊆B int_seg: {i..j-} uimplies: supposing a guard: {T} 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] false: False not: ¬A top: Top prop: less_than: a < b squash: T so_lambda: λ2x.t[x] sq_stable: SqStable(P)
Lemmas referenced :  sq_stable_wf all_wf list_wf int_seg_wf equal_wf set_wf int_formula_prop_less_lemma intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le length_wf int_seg_properties select_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin setEquality cumulativity hypothesisEquality applyEquality hypothesis because_Cache sqequalRule setElimination rename independent_isectElimination natural_numberEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination introduction imageMemberEquality baseClosed equalityTransitivity equalitySymmetry universeEquality functionEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].    ((\mforall{}x:T.  SqStable(P[x]))  {}\mRightarrow{}  (\mforall{}L:\{x:T|  P[x]\}    List.  (\mforall{}x\mmember{}L.P[x])))



Date html generated: 2016_05_14-AM-07_48_54
Last ObjectModification: 2016_01_15-AM-08_32_32

Theory : list_1


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