### Nuprl Lemma : bag-member-cons

`∀[T:Type]. ∀[x,u:T]. ∀[v:bag(T)].  uiff(x ↓∈ u.v;(x = u ∈ T) ↓∨ x ↓∈ v)`

Proof

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

Latex:
\mforall{}[T:Type].  \mforall{}[x,u:T].  \mforall{}[v:bag(T)].    uiff(x  \mdownarrow{}\mmember{}  u.v;(x  =  u)  \mdownarrow{}\mvee{}  x  \mdownarrow{}\mmember{}  v)

Date html generated: 2017_10_01-AM-08_54_02
Last ObjectModification: 2017_07_26-PM-04_35_50

Theory : bags

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