### Nuprl Lemma : list_in_mem_f_list

`∀T:Type. ∀as:T List.  (as ∈ {x:T| mem_f(T;x;as)}  List)`

Proof

Definitions occuring in Statement :  mem_f: `mem_f(T;a;bs)` list: `T List` all: `∀x:A. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` universe: `Type`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` or: `P ∨ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` mem_f: `mem_f(T;a;bs)` ycomb: `Y` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 istype-less_than list-cases list-subtype nil_wf subtype_rel_list l_member_wf mem_f_wf subtype_rel_sets null_nil_lemma btrue_wf member-implies-null-eq-bfalse btrue_neq_bfalse product_subtype_list colength-cons-not-zero colength_wf_list istype-le subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le le_wf list_ind_cons_lemma cons_wf equal_wf subtype_rel_list_set istype-nat list_wf istype-universe
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination independent_pairFormation universeIsType axiomEquality equalityTransitivity equalitySymmetry functionIsTypeImplies inhabitedIsType unionElimination because_Cache applyEquality setEquality setIsType promote_hyp hypothesis_subsumption productElimination equalityIstype dependent_set_memberEquality_alt instantiate applyLambdaEquality imageElimination baseApply closedConclusion baseClosed intEquality sqequalBase unionEquality inlFormation_alt unionIsType inrFormation_alt universeEquality

Latex:
\mforall{}T:Type.  \mforall{}as:T  List.    (as  \mmember{}  \{x:T|  mem\_f(T;x;as)\}    List)

Date html generated: 2019_10_16-PM-01_01_44
Last ObjectModification: 2019_06_20-PM-06_49_26

Theory : list_2

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