Nuprl Lemma : empty-bag-iff-no-member
∀[T:Type]. ∀[bs:bag(T)]. uiff(bs = {} ∈ bag(T);∀x:T. (¬x ↓∈ bs))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
empty-bag: {},
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
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,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag: bag(T),
quotient: x,y:A//B[x; y],
subtype_rel: A ⊆r B,
empty-bag: {},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
cons: [a / b],
bag-member: x ↓∈ bs,
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T
Lemmas referenced :
bag-member_wf,
equal-wf-T-base,
bag_wf,
all_wf,
not_wf,
bag-member-empty,
list-subtype-bag,
equal-wf-base,
list_wf,
permutation_wf,
quotient-member-eq,
permutation-equiv,
nil_wf,
permutation_inversion,
permutation-nil-iff,
equal_wf,
list-cases,
product_subtype_list,
cons_wf,
cons_member,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
because_Cache,
baseClosed,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
universeEquality,
hyp_replacement,
applyLambdaEquality,
independent_isectElimination,
pointwiseFunctionalityForEquality,
functionEquality,
pertypeElimination,
applyEquality,
productEquality,
rename,
unionElimination,
promote_hyp,
hypothesis_subsumption,
dependent_pairFormation,
inlFormation,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. uiff(bs = \{\};\mforall{}x:T. (\mneg{}x \mdownarrow{}\mmember{} bs))
Date html generated:
2017_10_01-AM-08_53_17
Last ObjectModification:
2017_07_26-PM-04_34_59
Theory : bags
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