Nuprl Lemma : list-equal-set
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[L,L':T List]. (L = L' ∈ ({x:T| P[x]} List)) supposing ((L = L' ∈ (T List)) and (∀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],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ
Lemmas referenced :
strong-subtype-equal-lists,
strong-subtype-set3,
strong-subtype-self,
list-set-type2,
equal_wf,
list_wf,
l_all_wf,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
sqequalRule,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[L,L':T List]. (L = L') supposing ((L = L') and (\mforall{}x\mmember{}L.P[x]))
Date html generated:
2016_05_14-AM-07_49_11
Last ObjectModification:
2015_12_26-PM-04_45_05
Theory : list_1
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