Nuprl Lemma : member_exists
∀[T:Type]. ∀L:T List. ∃x:T. (x ∈ L) supposing ¬(L = [] ∈ (T List))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
top: Top,
guard: {T},
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal-wf-T-base,
list_wf,
list-cases,
length-nil,
nil_wf,
product_subtype_list,
cons_wf,
length_of_nil_lemma,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
equal_wf,
not_wf,
select_wf,
le_wf,
less_than_wf,
length_wf,
sq_stable__le,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesis,
baseClosed,
rename,
unionElimination,
independent_functionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidEquality,
setElimination,
natural_numberEquality,
addEquality,
independent_pairFormation,
independent_isectElimination,
applyEquality,
intEquality,
because_Cache,
minusEquality,
dependent_pairFormation,
dependent_set_memberEquality,
productEquality,
imageMemberEquality,
imageElimination,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mexists{}x:T. (x \mmember{} L) supposing \mneg{}(L = [])
Date html generated:
2017_04_14-AM-08_37_09
Last ObjectModification:
2017_02_27-PM-03_28_58
Theory : list_0
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