Nuprl Lemma : dense-in-interval_wf
∀[I:Interval]. ∀[X:{a:ℝ| a ∈ I} ⟶ ℙ]. (dense-in-interval(I;X) ∈ ℙ)
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
i-member: r ∈ I,
interval: Interval,
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
exists: ∃x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
dense-in-interval: dense-in-interval(I;X),
member: t ∈ T,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a
Lemmas referenced :
interval_wf,
exists_wf,
rless_wf,
i-member_wf,
real_wf,
all_wf,
i-member-between,
rleq_weakening_rless
Rules used in proof :
isect_memberEquality,
universeEquality,
cumulativity,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
productElimination,
functionExtensionality,
applyEquality,
productEquality,
functionEquality,
because_Cache,
rename,
setElimination,
lambdaFormation,
lambdaEquality,
hypothesisEquality,
hypothesis,
setEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
dependent_set_memberEquality
Latex:
\mforall{}[I:Interval]. \mforall{}[X:\{a:\mBbbR{}| a \mmember{} I\} {}\mrightarrow{} \mBbbP{}]. (dense-in-interval(I;X) \mmember{} \mBbbP{})
Date html generated:
2017_10_03-AM-09_47_25
Last ObjectModification:
2017_08_30-PM-06_06_23
Theory : reals
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