Nuprl Lemma : totally-bounded-implies-nonvoid
∀[A:Set(ℝ)]. (totally-bounded(A) ⇒ (∃x:ℝ. (x ∈ A)))
Proof
Definitions occuring in Statement :
totally-bounded: totally-bounded(A),
rset-member: x ∈ A,
rset: Set(ℝ),
real: ℝ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
totally-bounded: totally-bounded(A),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
false: False,
guard: {T}
Lemmas referenced :
real_wf,
rless_wf,
int-to-real_wf,
nat_plus_wf,
int_seg_wf,
rset-member_wf,
rabs_wf,
rsub_wf,
rset_wf,
rless-int,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
istype-le,
istype-less_than
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
productIsType,
setElimination,
rename,
because_Cache,
applyEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
int_eqEquality
Latex:
\mforall{}[A:Set(\mBbbR{})]. (totally-bounded(A) {}\mRightarrow{} (\mexists{}x:\mBbbR{}. (x \mmember{} A)))
Date html generated:
2019_10_29-AM-10_43_31
Last ObjectModification:
2019_04_19-PM-06_11_28
Theory : reals
Home
Index