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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