Nuprl Lemma : nil_member
∀[T:Type]. ∀x:T. ((x ∈ []) ⇐⇒ False)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
nil: [],
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
false: False,
universe: Type
Definitions unfolded in proof :
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
false: False,
select: L[n],
member: t ∈ T,
uimplies: b supposing a,
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
cand: A c∧ B,
nat: ℕ,
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
length_of_nil_lemma,
stuck-spread,
base_wf,
less_than_transitivity1,
less_than_irreflexivity,
exists_wf,
nat_wf,
less_than_wf,
length_wf,
nil_wf,
equal_wf,
select_wf,
sq_stable__le,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
sqequalHypSubstitution,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
thin,
baseClosed,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
independent_functionElimination,
lambdaEquality,
productEquality,
because_Cache,
cumulativity,
imageMemberEquality,
imageElimination,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}x:T. ((x \mmember{} []) \mLeftarrow{}{}\mRightarrow{} False)
Date html generated:
2017_04_14-AM-08_37_14
Last ObjectModification:
2017_02_27-PM-03_28_55
Theory : list_0
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