Nuprl Lemma : property-from-l_member
∀[T:Type]. ∀x:T. ∀[P:T ⟶ ℙ]. ((∀x:T. SqStable(P[x])) ⇒ (∀d:{i:T| P[i]} List. ((x ∈ d) ⇒ P[x])))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
squash: ↓T,
sq_stable: SqStable(P)
Lemmas referenced :
sq_stable_wf,
all_wf,
list_wf,
subtype_rel_list,
l_member_wf,
l_member_settype
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_isectElimination,
applyEquality,
setEquality,
because_Cache,
sqequalRule,
lambdaEquality,
setElimination,
rename,
universeEquality,
functionEquality,
cumulativity,
imageMemberEquality,
baseClosed,
introduction,
imageElimination,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:T. SqStable(P[x])) {}\mRightarrow{} (\mforall{}d:\{i:T| P[i]\} List. ((x \mmember{} d) {}\mRightarrow{} P[x])))
Date html generated:
2016_05_14-AM-07_48_42
Last ObjectModification:
2016_01_15-AM-08_32_39
Theory : list_1
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