Nuprl Lemma : interval-connected
∀I:Interval. Connected({x:ℝ| x ∈ I} )
Proof
Definitions occuring in Statement :
connected: Connected(X),
i-member: r ∈ I,
interval: Interval,
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],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
uimplies: b supposing a,
top: Top,
subinterval: I ⊆ J ,
and: P ∧ Q,
cand: A c∧ B,
guard: {T}
Lemmas referenced :
all_wf,
real_wf,
i-member_wf,
or_wf,
exists_wf,
req_wf,
set_wf,
interval_wf,
closed-interval-connected,
rmin_wf,
rmax_wf,
rmin-rmax-subinterval,
sq_stable__i-member,
subtype_rel_dep_function,
rccint_wf,
subtype_rel_sets,
member_rccint_lemma,
subtype_rel_self,
sq_stable__rleq,
rmin-rleq,
rleq-rmax,
rleq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
setEquality,
hypothesis,
hypothesisEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
applyEquality,
functionExtensionality,
because_Cache,
dependent_set_memberEquality,
functionEquality,
universeEquality,
cumulativity,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
dependent_pairFormation,
productEquality
Latex:
\mforall{}I:Interval. Connected(\{x:\mBbbR{}| x \mmember{} I\} )
Date html generated:
2017_10_03-AM-10_12_16
Last ObjectModification:
2017_07_10-PM-05_40_57
Theory : reals
Home
Index