Nuprl Lemma : reals-connected
Connected(ℝ)
Proof
Definitions occuring in Statement :
connected: Connected(X),
real: ℝ
Definitions unfolded in proof :
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
so_apply: x[s],
member: t ∈ T,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
sq_exists: ∃x:{A| B[x]},
sq_stable: SqStable(P),
squash: ↓T,
cand: A c∧ B,
subtype_rel: A ⊆r B,
guard: {T},
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
real: ℝ,
uimplies: b supposing a,
connected: Connected(X),
false: False,
not: ¬A,
isl: isl(x),
outl: outl(x),
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isr: isr(x),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
sq_type: SQType(T)
Lemmas referenced :
req_wf,
bool_wf,
exists_wf,
or_wf,
all_wf,
real_wf,
assert_wf,
inhabited-covers-real-implies-ext,
connectedness-main-lemma-ext,
sq_stable__req,
nat_wf,
accelerate_wf,
less_than_wf,
subtype_rel_sets,
nat_plus_wf,
regular-int-seq_wf,
real-regular,
and_wf,
equal_wf,
isr_wf,
isl_wf,
btrue_neq_bfalse,
bfalse_wf,
assert_elim,
true_wf,
false_wf,
assert_of_bnot,
outr_wf,
bool_subtype_base,
subtype_base_sq,
btrue_wf,
squash_wf,
assert_functionality_wrt_uiff
Rules used in proof :
cut,
universeEquality,
cumulativity,
rename,
setElimination,
setEquality,
functionEquality,
because_Cache,
productElimination,
independent_functionElimination,
hypothesis,
hypothesisEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_functionElimination,
unionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
independent_pairFormation,
productEquality,
dependent_set_memberEquality,
natural_numberEquality,
intEquality,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
instantiate,
voidElimination,
levelHypothesis,
addLevel,
unionEquality,
inrEquality,
voidEquality,
inrFormation,
inlFormation
Latex:
Connected(\mBbbR{})
Date html generated:
2017_10_03-AM-10_11_42
Last ObjectModification:
2017_09_13-PM-03_29_27
Theory : reals
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