### Nuprl Lemma : bag-member-iff

`∀[T:Type]. ∀[bs:bag(T)]. ∀[x:T].  uiff(x ↓∈ bs;↓∃as:bag(T). (bs = ({x} + as) ∈ bag(T)))`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-append: `as + bs` single-bag: `{x}` bag: `bag(T)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` squash: `↓T` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` squash: `↓T` prop: `ℙ` bag-member: `x ↓∈ bs` so_lambda: `λ2x.t[x]` all: `∀x:A. B[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` single-bag: `{x}` bag-append: `as + bs` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` sq_stable: `SqStable(P)` implies: `P `` Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_or: `a ↓∨ b` or: `P ∨ Q`
Lemmas referenced :  bag-member_wf squash_wf exists_wf bag_wf equal_wf bag-append_wf single-bag_wf bag-member-iff-hd list-subtype-bag list_ind_cons_lemma list_ind_nil_lemma sq_stable__bag-member bag-member-append bag-member-single
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation hypothesis sqequalHypSubstitution imageElimination sqequalRule imageMemberEquality hypothesisEquality thin baseClosed extract_by_obid isectElimination cumulativity lambdaEquality dependent_functionElimination productElimination independent_pairEquality isect_memberEquality because_Cache equalityTransitivity equalitySymmetry universeEquality independent_isectElimination dependent_pairFormation applyEquality voidElimination voidEquality independent_functionElimination hyp_replacement Error :applyLambdaEquality,  inlFormation

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].  \mforall{}[x:T].    uiff(x  \mdownarrow{}\mmember{}  bs;\mdownarrow{}\mexists{}as:bag(T).  (bs  =  (\{x\}  +  as)))

Date html generated: 2016_10_25-AM-10_27_34
Last ObjectModification: 2016_07_12-AM-06_43_34

Theory : bags

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