### Nuprl Lemma : member-from-upto

`∀n,m:ℤ. ∀k:{x:ℤ| (n ≤ x) ∧ x < m} .  (k ∈ [n, m))`

Proof

Definitions occuring in Statement :  from-upto: `[n, m)` l_member: `(x ∈ l)` less_than: `a < b` le: `A ≤ B` all: `∀x:A. B[x]` and: `P ∧ Q` set: `{x:A| B[x]} ` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` member: `t ∈ T` and: `P ∧ Q` nat: `ℕ` uall: `∀[x:A]. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` cand: `A c∧ B` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  decidable__le subtract_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_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_subtract_lemma int_term_value_var_lemma int_formula_prop_wf le_wf length-from-upto lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int decidable__lt intformless_wf int_formula_prop_less_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot less_than_wf select-from-upto lelt_wf decidable__equal_int intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma length_wf from-upto_wf equal-wf-base-T select_wf nat_properties set_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation dependent_pairFormation setElimination thin rename cut sqequalHypSubstitution productElimination dependent_set_memberEquality because_Cache introduction extract_by_obid dependent_functionElimination natural_numberEquality isectElimination hypothesisEquality hypothesis unionElimination independent_isectElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll equalityElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity independent_functionElimination addEquality productEquality setEquality

Latex:
\mforall{}n,m:\mBbbZ{}.  \mforall{}k:\{x:\mBbbZ{}|  (n  \mleq{}  x)  \mwedge{}  x  <  m\}  .    (k  \mmember{}  [n,  m))

Date html generated: 2017_04_17-AM-07_55_16
Last ObjectModification: 2017_02_27-PM-04_26_35

Theory : list_1

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