### 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))`

Proof

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: `b 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: `b supposing a` bag-member: `x ↓∈ bs` squash: `↓T` prop: `ℙ` subtype_rel: `A ⊆r B` nat: `ℕ` 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: `P `` Q` bag: `bag(T)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` ge: `i ≥ j ` 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: `P `⇐⇒` Q` rev_implies: `P `` Q` le: `A ≤ B` less_than': `less_than'(a;b)` cand: `A 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

Latex:
\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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