Nuprl Lemma : l_member_set
∀[A:Type]. ∀[P:A ⟶ ℙ].  ∀L:A List. ∀x:A.  ((∀x∈L.P[x]) 
⇒ {(x ∈ L) 
⇒ (x ∈ L)})
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
, 
guard: {T}
, 
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 : 
guard: {T}
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
prop: ℙ
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
l_all_iff, 
l_member_wf, 
l_member-settype, 
list-subtype, 
subtype_rel_list_set, 
subtype_rel_self, 
l_all_wf, 
list_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
because_Cache, 
dependent_functionElimination, 
hypothesisEquality, 
lambdaEquality, 
applyEquality, 
setElimination, 
rename, 
hypothesis, 
setEquality, 
productElimination, 
independent_functionElimination, 
cumulativity, 
equalityTransitivity, 
equalitySymmetry, 
independent_isectElimination, 
dependent_set_memberEquality, 
instantiate, 
functionEquality, 
universeEquality
Latex:
\mforall{}[A:Type].  \mforall{}[P:A  {}\mrightarrow{}  \mBbbP{}].    \mforall{}L:A  List.  \mforall{}x:A.    ((\mforall{}x\mmember{}L.P[x])  {}\mRightarrow{}  \{(x  \mmember{}  L)  {}\mRightarrow{}  (x  \mmember{}  L)\})
Date html generated:
2019_06_20-PM-01_24_55
Last ObjectModification:
2018_08_24-PM-10_49_56
Theory : list_1
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