Nuprl Lemma : closed-interval-connected
∀u,v:ℝ. Connected({x:ℝ| x ∈ [u, v]} )
Proof
Definitions occuring in Statement :
connected: Connected(X),
rccint: [l, u],
i-member: r ∈ I,
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
connected: Connected(X),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
i-member: r ∈ I,
rccint: [l, u],
and: P ∧ Q,
member: t ∈ T,
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
or: P ∨ Q
Lemmas referenced :
reals-connected,
member_rccint_lemma,
sq_stable__rleq,
rleq_transitivity,
all_wf,
real_wf,
i-member_wf,
rccint_wf,
or_wf,
exists_wf,
req_wf,
set_wf,
interval-retraction_wf,
req_weakening,
sq_stable__req,
interval-retraction_functionality,
interval-retraction-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isect_memberFormation,
productElimination,
thin,
sqequalRule,
independent_pairFormation,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
setElimination,
rename,
isectElimination,
hypothesisEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
independent_isectElimination,
setEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
functionEquality,
universeEquality,
cumulativity,
dependent_pairFormation,
productEquality
Latex:
\mforall{}u,v:\mBbbR{}. Connected(\{x:\mBbbR{}| x \mmember{} [u, v]\} )
Date html generated:
2017_10_03-AM-10_11_59
Last ObjectModification:
2017_07_10-PM-05_32_02
Theory : reals
Home
Index