Nuprl Lemma : union-of-intervals-not-interval
¬(∀t:{t:ℝ| t ∈ [r(-1), r1]} . ((↓t ∈ [r(-1), r0]) ∨ (↓t ∈ [r0, r1])))
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
not: ¬A,
squash: ↓T,
or: P ∨ Q,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x],
false: False,
uall: ∀[x:A]. B[x],
prop: ℙ,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
uimplies: b supposing a,
top: Top,
cand: A c∧ B,
sq_stable: SqStable(P)
Lemmas referenced :
not-all-nonneg-or-nonpos,
real_wf,
i-member_wf,
rccint_wf,
int-to-real_wf,
squash_wf,
rless-cases,
rless-int,
rleq_weakening_rless,
rleq_wf,
member_rccint_lemma,
istype-void,
sq_stable__rleq
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
independent_functionElimination,
thin,
universeIsType,
hypothesis,
voidElimination,
sqequalRule,
functionIsType,
setIsType,
isectElimination,
minusEquality,
natural_numberEquality,
hypothesisEquality,
unionIsType,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
productElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
unionElimination,
inrFormation_alt,
independent_isectElimination,
inlFormation_alt,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
productIsType,
imageElimination
Latex:
\mneg{}(\mforall{}t:\{t:\mBbbR{}| t \mmember{} [r(-1), r1]\} . ((\mdownarrow{}t \mmember{} [r(-1), r0]) \mvee{} (\mdownarrow{}t \mmember{} [r0, r1])))
Date html generated:
2019_10_30-AM-07_20_07
Last ObjectModification:
2019_05_08-PM-11_33_09
Theory : reals
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