Nuprl Lemma : metric-subspace_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[A:Type]. (metric-subspace(X;d;A) ∈ ℙ)
Proof
Definitions occuring in Statement :
metric-subspace: metric-subspace(X;d;A),
metric: metric(X),
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
metric-subspace: metric-subspace(X;d;A),
prop: ℙ,
and: P ∧ Q,
member: t ∈ T,
uiff: uiff(P;Q),
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
metric_wf,
istype-universe,
strong-subtype_wf,
strong-subtype-iff-respects-equality,
all_wf,
meq_wf,
equal-wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
productEquality,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
instantiate,
universeEquality,
productElimination,
independent_isectElimination,
sqequalRule,
functionEquality,
because_Cache,
applyEquality,
lambdaFormation_alt,
lambdaEquality_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[A:Type]. (metric-subspace(X;d;A) \mmember{} \mBbbP{})
Date html generated:
2019_10_30-AM-06_30_25
Last ObjectModification:
2019_10_02-AM-10_05_24
Theory : reals
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