Nuprl Lemma : member_iff_sublist

`∀[T:Type]. ∀x:T. ∀L:T List.  ((x ∈ L) `⇐⇒` [x] ⊆ L)`

Proof

Definitions occuring in Statement :  sublist: `L1 ⊆ L2` l_member: `(x ∈ l)` cons: `[a / b]` nil: `[]` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` universe: `Type`
Definitions unfolded in proof :  sublist: `L1 ⊆ L2` l_member: `(x ∈ l)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` top: `Top` exists: `∃x:A. B[x]` cand: `A c∧ B` prop: `ℙ` so_lambda: `λ2x.t[x]` nat: `ℕ` uimplies: `b supposing a` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` so_apply: `x[s]` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` increasing: `increasing(f;k)` subtract: `n - m` sq_type: `SQType(T)` select: `L[n]` cons: `[a / b]` less_than: `a < b` squash: `↓T` true: `True`
Lemmas referenced :  length_of_cons_lemma length_of_nil_lemma exists_wf nat_wf less_than_wf length_wf equal_wf select_wf nat_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 int_seg_wf cons_wf nil_wf increasing_wf length_wf_nat all_wf int_seg_properties decidable__lt intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma non_neg_length lelt_wf list_wf false_wf le_wf length-singleton decidable__equal_int intformeq_wf int_formula_prop_eq_lemma subtype_base_sq set_subtype_base int_subtype_base
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isect_memberEquality voidElimination voidEquality hypothesis productElimination isectElimination lambdaEquality productEquality setElimination rename because_Cache cumulativity hypothesisEquality independent_isectElimination natural_numberEquality unionElimination dependent_pairFormation int_eqEquality intEquality computeAll functionEquality functionExtensionality applyEquality addEquality independent_functionElimination dependent_set_memberEquality equalityTransitivity equalitySymmetry applyLambdaEquality universeEquality instantiate imageMemberEquality baseClosed

Latex:
\mforall{}[T:Type].  \mforall{}x:T.  \mforall{}L:T  List.    ((x  \mmember{}  L)  \mLeftarrow{}{}\mRightarrow{}  [x]  \msubseteq{}  L)

Date html generated: 2017_04_14-AM-09_29_39
Last ObjectModification: 2017_02_27-PM-04_02_02

Theory : list_1

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