Nuprl Lemma : bag-member-empty
∀[T:Type]. ∀[x:T]. False supposing x ↓∈ {}
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
empty-bag: {},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
false: False,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
false: False,
bag-member: x ↓∈ bs,
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
all: ∀x:A. B[x],
or: P ∨ Q,
bag-size: #(bs),
not: ¬A,
implies: P ⇒ Q,
cons: [a / b],
top: Top,
uiff: uiff(P;Q),
bag-null: bag-null(bs),
null: null(as),
bfalse: ff,
assert: ↑b,
ifthenelse: if b then t else f fi
Lemmas referenced :
bag_size_empty_lemma,
equal-wf-T-base,
bag-size_wf,
nat_wf,
bag-member_wf,
empty-bag_wf,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
product_subtype_list,
length_of_cons_lemma,
assert-bag-null
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
hypothesis,
sqequalRule,
extract_by_obid,
natural_numberEquality,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
isectElimination,
intEquality,
cumulativity,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
baseClosed,
because_Cache,
isect_memberEquality,
equalityTransitivity,
voidElimination,
universeEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
independent_functionElimination,
promote_hyp,
hypothesis_subsumption,
voidEquality,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. False supposing x \mdownarrow{}\mmember{} \{\}
Date html generated:
2016_10_25-AM-10_26_58
Last ObjectModification:
2016_07_12-AM-06_43_04
Theory : bags
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