### Nuprl Lemma : bag-size-bag-member

`∀[T:Type]. ∀[bs:bag(T)].  (0 < #(bs) `⇐⇒` ↓∃b:T. b ↓∈ bs)`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-size: `#(bs)` bag: `bag(T)` less_than: `a < b` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` squash: `↓T` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` squash: `↓T` exists: `∃x:A. B[x]` prop: `ℙ` subtype_rel: `A ⊆r B` bag-size: `#(bs)` bag-member: `x ↓∈ bs` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` all: `∀x:A. B[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` rev_implies: `P `` Q` bag: `bag(T)` quotient: `x,y:A//B[x; y]` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` top: `Top` decidable: `Dec(P)` uiff: `uiff(P;Q)` subtract: `n - m` true: `True`
Lemmas referenced :  bag_to_squash_list less_than_wf bag-size_wf select_wf false_wf select_member lelt_wf length_wf l_member_wf list-subtype-bag equal_wf bag_wf squash_wf exists_wf list_wf bag-member_wf nat_wf member-less_than member-permutation list-cases length_of_nil_lemma nil_member product_subtype_list length_of_cons_lemma length_wf_nat decidable__lt not-lt-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-zero le-add-cancel member_wf permutation_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality imageElimination productElimination promote_hyp hypothesis equalitySymmetry hyp_replacement applyLambdaEquality natural_numberEquality cumulativity applyEquality because_Cache sqequalRule rename dependent_pairFormation independent_isectElimination dependent_functionElimination dependent_set_memberEquality lambdaEquality productEquality imageMemberEquality baseClosed setElimination independent_pairEquality isect_memberEquality universeEquality pertypeElimination independent_functionElimination unionElimination voidElimination hypothesis_subsumption voidEquality addEquality intEquality minusEquality equalityTransitivity

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].    (0  <  \#(bs)  \mLeftarrow{}{}\mRightarrow{}  \mdownarrow{}\mexists{}b:T.  b  \mdownarrow{}\mmember{}  bs)

Date html generated: 2017_10_01-AM-08_54_49
Last ObjectModification: 2017_07_26-PM-04_36_40

Theory : bags

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