Nuprl Lemma : member-implies-null-eq-bfalse
∀[T:Type]. ∀[L:T List]. ∀[x:T]. null(L) = ff supposing (x ∈ L)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
null: null(as),
list: T List,
bfalse: ff,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
or: P ∨ Q,
cons: [a / b],
top: Top,
prop: ℙ,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
iff_imp_equal_bool,
null_wf,
bfalse_wf,
null_nil_lemma,
btrue_wf,
list-cases,
nil_member,
product_subtype_list,
null_cons_lemma,
and_wf,
equal_wf,
list_wf,
btrue_neq_bfalse,
nil_wf,
false_wf,
assert_of_null,
assert_wf,
iff_wf,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
sqequalRule,
dependent_functionElimination,
unionElimination,
productElimination,
independent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
addLevel,
impliesFunctionality,
axiomEquality,
because_Cache,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[x:T]. null(L) = ff supposing (x \mmember{} L)
Date html generated:
2016_05_14-AM-06_38_46
Last ObjectModification:
2015_12_26-PM-00_32_12
Theory : list_0
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