### Nuprl Lemma : member-exists2

`∀[T:Type]. ∀L:T List. (∃x:T. (x ∈ L) `⇐⇒` 0 < ||L||)`

Proof

Definitions occuring in Statement :  l_member: `(x ∈ l)` length: `||as||` list: `T List` less_than: `a < b` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` ge: `i ≥ j ` member: `t ∈ T` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` rev_implies: `P `` Q` less_than: `a < b` squash: `↓T` uiff: `uiff(P;Q)` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  l_member_wf exists_wf member-exists iff_wf nil_wf list_wf equal_wf not_wf length_of_not_nil less_than_wf less_than'_wf decidable__le ge_wf int_formula_prop_wf int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformle_wf itermVar_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt length_wf decidable__lt
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut independent_pairFormation sqequalHypSubstitution lemma_by_obid dependent_functionElimination thin natural_numberEquality isectElimination hypothesisEquality hypothesis unionElimination productElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule computeAll introduction imageElimination independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry addLevel impliesFunctionality independent_functionElimination universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  (\mexists{}x:T.  (x  \mmember{}  L)  \mLeftarrow{}{}\mRightarrow{}  0  <  ||L||)

Date html generated: 2016_05_14-PM-01_30_26
Last ObjectModification: 2016_01_15-AM-08_27_37

Theory : list_1

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