Nuprl Lemma : i-member-between
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (∀r:ℝ. ((a ≤ r) ⇒ (r ≤ b) ⇒ (r ∈ I))))
Proof
Definitions occuring in Statement :
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
interval: Interval,
i-member: r ∈ I,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
uimplies: b supposing a,
true: True,
prop: ℙ
Lemmas referenced :
rleq_transitivity,
rless_transitivity1,
rless_transitivity2,
rleq_wf,
real_wf,
i-member_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
unionElimination,
sqequalRule,
cut,
hypothesis,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality
Latex:
\mforall{}I:Interval. \mforall{}a,b:\mBbbR{}. ((a \mmember{} I) {}\mRightarrow{} (b \mmember{} I) {}\mRightarrow{} (\mforall{}r:\mBbbR{}. ((a \mleq{} r) {}\mRightarrow{} (r \mleq{} b) {}\mRightarrow{} (r \mmember{} I))))
Date html generated:
2016_05_18-AM-08_41_16
Last ObjectModification:
2015_12_27-PM-11_50_41
Theory : reals
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