Nuprl Lemma : list-set-type2
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ ℙ]. L ∈ {x:T| P[x]} List supposing (∀x∈L.P[x])
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
guard: {T}
Lemmas referenced :
list-set-type,
l_all_wf,
l_member_wf,
list_wf,
l_all_iff,
subtype_rel_list_set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
applyEquality,
sqequalRule,
axiomEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
isect_memberEquality,
because_Cache,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
lambdaFormation
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. L \mmember{} \{x:T| P[x]\} List supposing (\mforall{}x\mmember{}L.P[x])
Date html generated:
2016_05_14-AM-06_41_11
Last ObjectModification:
2016_03_15-PM-04_40_08
Theory : list_0
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