Nuprl Lemma : l_member-set2

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


Proof




Definitions occuring in Statement :  l_member: (x ∈ l) list: List uall: [x:A]. B[x] prop: so_apply: x[s] all: x:A. B[x] implies:  Q member: t ∈ T set: {x:A| B[x]}  function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B so_apply: x[s] guard: {T} so_lambda: λ2x.t[x] or: P ∨ Q iff: ⇐⇒ Q cons: [a b] colength: colength(L) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] decidable: Dec(P) nil: [] it: sq_type: SQType(T) less_than: a < b squash: T less_than': less_than'(a;b)
Lemmas referenced :  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 l_member_wf subtype_rel_list equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity set_wf list-cases nil_member nil_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int cons_member and_wf cons_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry cumulativity applyEquality setEquality functionExtensionality because_Cache unionElimination productElimination promote_hyp hypothesis_subsumption applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination hyp_replacement functionEquality universeEquality

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



Date html generated: 2017_04_17-AM-07_25_18
Last ObjectModification: 2017_02_27-PM-04_04_28

Theory : list_1


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