### Nuprl Lemma : bag-append-is-single-iff

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

Proof

Definitions occuring in Statement :  bag-append: `as + bs` single-bag: `{x}` empty-bag: `{}` bag: `bag(T)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` squash: `↓T` or: `P ∨ Q` and: `P ∧ Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` squash: `↓T` prop: `ℙ` or: `P ∨ Q` subtype_rel: `A ⊆r B` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q`
Lemmas referenced :  equal_wf bag_wf bag-append_wf single-bag_wf squash_wf or_wf equal-wf-T-base bag-append-is-single bag-append-empty bag-subtype-list true_wf bag-append-comm iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation independent_pairFormation hypothesis sqequalHypSubstitution imageElimination sqequalRule imageMemberEquality hypothesisEquality thin baseClosed extract_by_obid isectElimination cumulativity dependent_functionElimination productEquality lambdaEquality productElimination independent_pairEquality isect_memberEquality equalityTransitivity equalitySymmetry axiomEquality because_Cache universeEquality independent_isectElimination unionElimination equalityElimination applyEquality hyp_replacement applyLambdaEquality natural_numberEquality independent_functionElimination

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

Date html generated: 2017_10_01-AM-08_46_51
Last ObjectModification: 2017_07_26-PM-04_31_32

Theory : bags

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