Nuprl Lemma : member-approx-type
∀[T:Type]. ∀x:Base. uiff(x ∈ approx-type(T);↓∃t:Base. ((x ≤ t) ∧ (t ∈ T)))
Proof
Definitions occuring in Statement :
approx-type: approx-type(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
member: t ∈ T,
base: Base,
universe: Type,
sqle: s ≤ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
exists: ∃x:A. B[x],
approx-type: approx-type(T),
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
approx-per: approx-per(T;x;y),
cand: A c∧ B,
implies: P ⇒ Q
Lemmas referenced :
approx-type_wf,
squash_wf,
base_wf,
sqle_wf_base,
equal-wf-base,
istype-base,
istype-universe,
approx-per_wf,
istype-sqle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
Error :equalityIsType4,
Error :universeIsType,
extract_by_obid,
isectElimination,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
Error :lambdaEquality_alt,
dependent_functionElimination,
productElimination,
independent_pairEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
instantiate,
universeEquality,
pertypeElimination,
promote_hyp,
applyEquality,
pertypeMemberEquality,
Error :dependent_pairFormation_alt,
Error :productIsType,
Error :equalityIsType2
Latex:
\mforall{}[T:Type]. \mforall{}x:Base. uiff(x \mmember{} approx-type(T);\mdownarrow{}\mexists{}t:Base. ((x \mleq{} t) \mwedge{} (t \mmember{} T)))
Date html generated:
2019_06_20-PM-00_34_59
Last ObjectModification:
2018_11_20-PM-03_29_38
Theory : partial_1
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