Nuprl Lemma : i-closed-closed
∀I:Interval. (i-closed(I) ⇒ closed-rset(λx.(x ∈ I)))
Proof
Definitions occuring in Statement :
i-member: r ∈ I,
i-closed: i-closed(I),
interval: Interval,
closed-rset: closed-rset(A),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
closed-rset: closed-rset(A),
member-closure: y ∈ closure(A),
exists: ∃x:A. B[x],
and: P ∧ Q,
interval: Interval,
i-member: r ∈ I,
i-closed: i-closed(I),
isl: isl(x),
outl: outl(x),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
bor: p ∨bq,
bfalse: ff,
assert: ↑b,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
false: False,
real: ℝ,
prop: ℙ,
true: True,
guard: {T}
Lemmas referenced :
rleq-limit,
nat_wf,
constant-limit,
req_weakening,
regular-int-seq_wf,
member-closure_wf,
i-member_wf,
real_wf,
i-closed_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
unionElimination,
cut,
rename,
lemma_by_obid,
isectElimination,
lambdaEquality,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
voidElimination,
setElimination,
dependent_set_memberEquality,
natural_numberEquality
Latex:
\mforall{}I:Interval. (i-closed(I) {}\mRightarrow{} closed-rset(\mlambda{}x.(x \mmember{} I)))
Date html generated:
2016_05_18-AM-09_20_56
Last ObjectModification:
2015_12_27-PM-11_23_23
Theory : reals
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