Nuprl Lemma : bag-member-sv-bag-only

[T:Type]. ∀[b:bag(T)].  (sv-bag-only(b) ↓∈ b) supposing (0 < #(b) and single-valued-bag(b;T))


Definitions occuring in Statement :  sv-bag-only: sv-bag-only(b) single-valued-bag: single-valued-bag(b;T) bag-member: x ↓∈ bs bag-size: #(bs) bag: bag(T) less_than: a < b uimplies: supposing a uall: [x:A]. B[x] natural_number: $n universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a bag-member: x ↓∈ bs squash: T prop: subtype_rel: A ⊆B nat: exists: x:A. B[x] and: P ∧ Q sv-bag-only: sv-bag-only(b) bag-size: #(bs) single-valued-bag: single-valued-bag(b;T) implies:  Q bag: bag(T) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] all: x:A. B[x] ge: i ≥  guard: {T} int_seg: {i..j-} lelt: i ≤ j < k decidable: Dec(P) or: P ∨ Q less_than: a < b satisfiable_int_formula: satisfiable_int_formula(fmla) false: False not: ¬A top: Top so_lambda: λ2x.t[x] so_apply: x[s] true: True iff: ⇐⇒ Q rev_implies:  Q le: A ≤ B less_than': less_than'(a;b) cand: c∧ B l_member: (x ∈ l)
Lemmas referenced :  less_than_wf bag-size_wf nat_wf single-valued-bag_wf bag_wf mklist_wf sv-bag-only_wf int_seg_wf equal_wf list-subtype-bag l_member_wf bag_to_squash_list quotient-member-eq list_wf permutation_wf permutation-equiv length_wf_nat hd_wf int_seg_properties length_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_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_less_lemma int_formula_prop_wf all_wf bag-member_wf permutation_weakening list_extensionality mklist_length mklist_select squash_wf true_wf nat_properties lelt_wf select_wf iff_weakening_equal select0 false_wf select_member le_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution imageElimination hypothesis imageMemberEquality hypothesisEquality thin baseClosed extract_by_obid isectElimination natural_numberEquality cumulativity applyEquality lambdaEquality setElimination rename isect_memberEquality because_Cache equalityTransitivity equalitySymmetry universeEquality dependent_pairFormation independent_isectElimination independent_pairFormation productEquality productElimination lambdaFormation dependent_functionElimination unionElimination int_eqEquality intEquality voidElimination voidEquality computeAll independent_functionElimination functionEquality hyp_replacement applyLambdaEquality dependent_set_memberEquality

\mforall{}[T:Type].  \mforall{}[b:bag(T)].    (sv-bag-only(b)  \mdownarrow{}\mmember{}  b)  supposing  (0  <  \#(b)  and  single-valued-bag(b;T))

Date html generated: 2017_10_01-AM-08_55_41
Last ObjectModification: 2017_07_26-PM-04_37_49

Theory : bags

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