### Nuprl Lemma : list-cardinality-le

`∀[T:Type]. ∀L:T List. ((∀x:T. (x ∈ L)) `` |T| ≤ ||L||)`

Proof

Definitions occuring in Statement :  cardinality-le: `|T| ≤ n` l_member: `(x ∈ l)` length: `||as||` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` universe: `Type`
Definitions unfolded in proof :  cardinality-le: `|T| ≤ n` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` prop: `ℙ` less_than: `a < b` squash: `↓T` surject: `Surj(A;B;f)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` l_member: `(x ∈ l)` nat: `ℕ` le: `A ≤ B` cand: `A c∧ B` ge: `i ≥ j `
Lemmas referenced :  nat_properties equal_wf lelt_wf list_wf l_member_wf all_wf surject_wf int_seg_wf int_formula_prop_less_lemma intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le length_wf int_seg_properties select_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation dependent_pairFormation lambdaEquality cut lemma_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality setElimination rename hypothesis independent_isectElimination natural_numberEquality productElimination dependent_functionElimination unionElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll because_Cache imageElimination universeEquality dependent_set_memberEquality equalitySymmetry

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

Date html generated: 2016_05_14-PM-01_51_46
Last ObjectModification: 2016_01_15-AM-08_15_56

Theory : list_1

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