Nuprl Lemma : equal-empty-bag
∀[T:Type]. ∀x:bag(T). ((x = {} ∈ bag(T)) ⇒ (x ~ {}))
Proof
Definitions occuring in Statement :
empty-bag: {},
bag: bag(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
empty-bag: {},
or: P ∨ Q,
cons: [a / b],
and: P ∧ Q,
top: Top,
not: ¬A,
false: False
Lemmas referenced :
bag-subtype-list,
list_wf,
top_wf,
equal_wf,
equal-wf-T-base,
bag_wf,
list-cases,
product_subtype_list,
null_nil_lemma,
btrue_wf,
and_wf,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesisEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
isectElimination,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
equalityTransitivity,
independent_functionElimination,
cumulativity,
baseClosed,
lambdaEquality,
sqequalAxiom,
because_Cache,
universeEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
rename,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}x:bag(T). ((x = \{\}) {}\mRightarrow{} (x \msim{} \{\}))
Date html generated:
2017_10_01-AM-08_44_55
Last ObjectModification:
2017_07_26-PM-04_30_25
Theory : bags
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