### 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`
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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