Nuprl Lemma : nil-iff-no-member
∀[T:Type]. ∀[L:T List]. uiff(L = [] ∈ (T List);∀[x:T]. (¬(x ∈ L)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
nil: [],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q
Lemmas referenced :
list_induction,
uiff_wf,
equal_wf,
list_wf,
nil_wf,
uall_wf,
not_wf,
l_member_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
and_wf,
null_wf3,
subtype_rel_list,
top_wf,
null_cons_lemma,
bfalse_wf,
cons_wf,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
independent_functionElimination,
independent_pairFormation,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
voidElimination,
dependent_functionElimination,
because_Cache,
isect_memberEquality,
rename,
productElimination,
dependent_set_memberEquality,
applyEquality,
setElimination,
voidEquality,
setEquality,
independent_pairEquality,
axiomEquality,
universeEquality,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. uiff(L = [];\mforall{}[x:T]. (\mneg{}(x \mmember{} L)))
Date html generated:
2016_05_15-PM-03_57_26
Last ObjectModification:
2015_12_27-PM-03_07_58
Theory : general
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