Nuprl Lemma : l_exists_iff
∀[T:Type]. ∀L:T List. ∀[P:{x:T| (x ∈ L)} ⟶ ℙ]. ((∃x∈L. P[x]) ⇐⇒ ∃x:T. ((x ∈ L) ∧ P[x]))
Proof
Definitions occuring in Statement :
l_exists: (∃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],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
l_exists: (∃x∈L. P[x]),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
cand: A c∧ B,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
label: ...$L... t,
guard: {T}
Lemmas referenced :
exists_wf,
int_seg_wf,
length_wf,
l_member_wf,
select_wf,
list-subtype,
sq_stable__le,
list_wf,
select_member,
lelt_wf,
less_than_wf,
equal_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productEquality,
dependent_set_memberEquality,
universeEquality,
functionEquality,
dependent_pairFormation,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. ((\mexists{}x\mmember{}L. P[x]) \mLeftarrow{}{}\mRightarrow{} \mexists{}x:T. ((x \mmember{} L) \mwedge{} P[x]))
Date html generated:
2017_04_14-AM-08_40_11
Last ObjectModification:
2017_02_27-PM-03_30_49
Theory : list_0
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