Nuprl Lemma : not-l_exists
∀[T:Type]. ∀L:T List. ∀[P:{x:T| (x ∈ L)} ⟶ ℙ]. (¬(∃x∈L. P[x]) ⇐⇒ (∀x∈L.¬P[x]))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
exists: ∃x:A. B[x],
guard: {T},
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
l_all: (∀x∈L.P[x]),
cand: A c∧ B
Lemmas referenced :
list_wf,
length_wf,
int_seg_wf,
sq_stable__le,
list-subtype,
select_wf,
l_all_wf,
l_exists_wf,
l_all_iff,
l_exists_iff,
iff_wf,
exists_wf,
not_over_exists,
not_wf,
all_wf,
l_member_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
lambdaFormation,
thin,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
applyEquality,
hypothesisEquality,
dependent_set_memberEquality,
lemma_by_obid,
isectElimination,
cumulativity,
sqequalRule,
lambdaEquality,
productEquality,
productElimination,
functionEquality,
addLevel,
impliesFunctionality,
independent_isectElimination,
universeEquality,
dependent_functionElimination,
setEquality,
impliesLevelFunctionality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
natural_numberEquality,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberFormation,
independent_pairEquality,
isect_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. (\mneg{}(\mexists{}x\mmember{}L. P[x]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x\mmember{}L.\mneg{}P[x]))
Date html generated:
2016_05_14-AM-06_40_52
Last ObjectModification:
2016_01_14-PM-08_20_13
Theory : list_0
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