Nuprl Lemma : not-not-l_all-shift
∀[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_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
or: P ∨ Q,
false: False,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s]
Lemmas referenced :
not-not-l_all-xmiddle,
int_seg_wf,
length_wf,
l_all_wf,
not_wf,
l_member_wf,
istype-void,
list_wf,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
voidElimination,
universeIsType,
natural_numberEquality,
sqequalRule,
lambdaEquality_alt,
unionEquality,
applyEquality,
setIsType,
functionIsType,
universeEquality,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x\mmember{}L.\mneg{}\mneg{}P[x]) {}\mRightarrow{} (\mneg{}\mneg{}(\mforall{}x\mmember{}L.P[x])))
Date html generated:
2020_05_19-PM-09_37_17
Last ObjectModification:
2019_11_04-PM-01_47_06
Theory : list_0
Home
Index