Nuprl Lemma : bag-void-sq-empty
∀[T:Type]. ∀x:bag(T). x ~ {} supposing ¬T
Proof
Definitions occuring in Statement :
empty-bag: {},
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
or: P ∨ Q,
nil: [],
it: ⋅,
empty-bag: {},
cons: [a / b],
not: ¬A,
false: False,
prop: ℙ
Lemmas referenced :
equal-empty-bag,
list-cases,
product_subtype_list,
empty-bag_wf,
list_wf,
permutation_wf,
istype-void,
bag_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
pointwiseFunctionalityForEquality,
because_Cache,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
productElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
unionElimination,
hypothesis_subsumption,
voidElimination,
equalityIstype,
productIsType,
universeIsType,
sqequalBase,
axiomSqEquality,
functionIsType,
lambdaEquality_alt,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}x:bag(T). x \msim{} \{\} supposing \mneg{}T
Date html generated:
2019_10_15-AM-10_59_53
Last ObjectModification:
2019_08_15-AM-10_32_40
Theory : bags
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