### Nuprl Lemma : bag-member-not-bag-null

`∀[T:Type]. ∀[bs:bag(T)].  uiff(↓∃x:T. x ↓∈ bs;¬↑bag-null(bs))`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-null: `bag-null(bs)` bag: `bag(T)` assert: `↑b` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` not: `¬A` squash: `↓T` universe: `Type`
Definitions unfolded in proof :  uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` member: `t ∈ T` not: `¬A` implies: `P `` Q` false: `False` squash: `↓T` exists: `∃x:A. B[x]` uall: `∀[x:A]. B[x]` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bag-null: `bag-null(bs)` bag-member: `x ↓∈ bs` cand: `A c∧ B`
Lemmas referenced :  assert-bag-null bag-member_wf squash_wf true_wf iff_weakening_equal bag-member-empty-iff assert_wf bag-null_wf exists_wf bag_to_squash_list not_wf assert_of_null equal-wf-T-base list_wf member_exists list-subtype-bag equal_wf bag_wf l_member_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity independent_pairFormation isect_memberFormation introduction cut lambdaFormation thin sqequalHypSubstitution imageElimination productElimination extract_by_obid isectElimination hypothesisEquality hypothesis independent_isectElimination applyEquality lambdaEquality equalityTransitivity equalitySymmetry because_Cache natural_numberEquality sqequalRule imageMemberEquality baseClosed universeEquality independent_functionElimination voidElimination dependent_functionElimination cumulativity promote_hyp hyp_replacement applyLambdaEquality rename dependent_pairFormation productEquality independent_pairEquality isect_memberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].    uiff(\mdownarrow{}\mexists{}x:T.  x  \mdownarrow{}\mmember{}  bs;\mneg{}\muparrow{}bag-null(bs))

Date html generated: 2017_10_01-AM-08_54_32
Last ObjectModification: 2017_07_26-PM-04_36_21

Theory : bags

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