Nuprl Lemma : reals-uncountable
∀z:ℕ ⟶ ℝ. ∀x,y:ℝ. ((x < y) ⇒ (∃u:ℝ. ((x ≤ u) ∧ (u ≤ y) ∧ (∀n:ℕ. u ≠ z n))))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rleq: x ≤ y,
rless: x < y,
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
cand: A c∧ B,
sq_type: SQType(T),
subtract: n - m,
converges-to: lim n→∞.x[n] = y,
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
true: True,
rge: x ≥ y,
rbetween: x≤y≤z
Lemmas referenced :
cantor-lemma2,
primrec_wf,
real_wf,
rless_wf,
pi1_wf_top,
pi2_wf,
subtype_rel_dep_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
set_wf,
subtype_rel_product,
top_wf,
equal_wf,
primrec0_lemma,
member_wf,
le_wf,
all_wf,
rleq_wf,
nat_properties,
sq_stable__less_than,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
or_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
exists_wf,
primrec-unroll,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
add-subtract-cancel,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
less_than_wf,
primrec-wf2,
rleq_weakening_equal,
add-zero,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
rless_transitivity2,
rless_transitivity1,
add-associates,
add-swap,
add-commutes,
zero-add,
rleq_transitivity,
trivial-int-eq1,
common-limit-squeeze,
rleq_weakening_rless,
nat_plus_wf,
rabs_wf,
rleq_functionality,
rabs-of-nonneg,
req_weakening,
radd-preserves-rleq,
radd_wf,
rminus_wf,
radd_functionality,
rminus-zero,
req_inversion,
radd-assoc,
radd_comm,
req_transitivity,
radd-ac,
radd-rminus-assoc,
radd-zero-both,
subtract-add-cancel,
req_wf,
req_functionality,
rleq-int-fractions,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add_functionality_wrt_le,
le-add-cancel,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq_functionality_wrt_implies,
rleq_weakening,
rneq_wf,
rleq-limit,
constant-limit,
limit-shift
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
setEquality,
productEquality,
hypothesis,
because_Cache,
sqequalRule,
dependent_set_memberEquality,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaEquality,
applyEquality,
functionEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionExtensionality,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
int_eqEquality,
intEquality,
computeAll,
inrFormation,
equalityElimination,
impliesFunctionality,
instantiate,
cumulativity,
hyp_replacement,
dependent_set_memberFormation,
minusEquality,
multiplyEquality,
inlFormation
Latex:
\mforall{}z:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}x,y:\mBbbR{}. ((x < y) {}\mRightarrow{} (\mexists{}u:\mBbbR{}. ((x \mleq{} u) \mwedge{} (u \mleq{} y) \mwedge{} (\mforall{}n:\mBbbN{}. u \mneq{} z n))))
Date html generated:
2017_10_03-AM-09_12_16
Last ObjectModification:
2017_07_28-AM-07_43_02
Theory : reals
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