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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