### Nuprl Lemma : no_repeats-l_member!

`∀[T:Type]. ∀l:T List. ∀x:T. ((x ∈ l) `⇐⇒` (x ∈! l)) supposing no_repeats(T;l)`

Proof

Definitions occuring in Statement :  l_member!: `(x ∈! l)` no_repeats: `no_repeats(T;l)` l_member: `(x ∈ l)` list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` cand: `A c∧ B` l_member!: `(x ∈! l)` nat: `ℕ` no_repeats: `no_repeats(T;l)` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` squash: `↓T` guard: `{T}` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_stable: `SqStable(P)` ge: `i ≥ j ` rev_implies: `P `` Q`
Lemmas referenced :  no_repeats_witness decidable__equal_int satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf equal-wf-base set_subtype_base le_wf int_subtype_base sq_stable__le decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma equal_wf select_wf nat_properties less_than_wf length_wf nat_wf all_wf l_member_wf l_member!_wf no_repeats_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_functionElimination hypothesis rename independent_pairFormation productElimination dependent_pairFormation dependent_set_memberEquality dependent_functionElimination setElimination because_Cache unionElimination independent_isectElimination applyLambdaEquality sqequalRule imageMemberEquality baseClosed imageElimination natural_numberEquality lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll applyEquality equalityTransitivity equalitySymmetry cumulativity productEquality functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}l:T  List.  \mforall{}x:T.  ((x  \mmember{}  l)  \mLeftarrow{}{}\mRightarrow{}  (x  \mmember{}!  l))  supposing  no\_repeats(T;l)

Date html generated: 2017_04_17-AM-07_29_12
Last ObjectModification: 2017_02_27-PM-04_07_36

Theory : list_1

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