Nuprl Lemma : l_all_not
∀[T:Type]. ∀L:T List. ∀P:T ⟶ 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]),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
list_wf,
not_wf,
all_wf,
sq_stable__le,
select_wf,
length_wf,
int_seg_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
thin,
sqequalHypSubstitution,
productElimination,
because_Cache,
hypothesis,
independent_functionElimination,
voidElimination,
lemma_by_obid,
isectElimination,
natural_numberEquality,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
setElimination,
rename,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_pairEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:T {}\mrightarrow{} \mBbbP{}. ((\mforall{}x\mmember{}L.\mneg{}P[x]) \mLeftarrow{}{}\mRightarrow{} \mneg{}(\mexists{}x\mmember{}L. P[x]))
Date html generated:
2016_05_14-AM-06_40_31
Last ObjectModification:
2016_01_14-PM-08_20_36
Theory : list_0
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