Nuprl Lemma : rationals-dense

`∀x:ℝ. ∀y:{y:ℝ| x < y} .  ∃n:ℕ+. ∃m:ℤ. ((x < (r(m)/r(n))) ∧ ((r(m)/r(n)) < y))`

Proof

Definitions occuring in Statement :  rdiv: `(x/y)` rless: `x < y` int-to-real: `r(n)` real: `ℝ` nat_plus: `ℕ+` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` and: `P ∧ Q` set: `{x:A| B[x]} ` int: `ℤ`
Definitions unfolded in proof :  top: `Top` not: `¬A` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` decidable: `Dec(P)` implies: `P `` Q` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` or: `P ∨ Q` guard: `{T}` rneq: `x ≠ y` and: `P ∧ Q` true: `True` less_than': `less_than'(a;b)` squash: `↓T` less_than: `a < b` exists: `∃x:A. B[x]` nat_plus: `ℕ+` uimplies: `b supposing a` sq_exists: `∃x:A [B[x]]` rless: `x < y` subtype_rel: `A ⊆r B` so_apply: `x[s]` prop: `ℙ` real: `ℝ` so_lambda: `λ2x.t[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` rational-approx: `(x within 1/n)` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` sq_type: `SQType(T)` uiff: `uiff(P;Q)` rge: `x ≥ y` rdiv: `(x/y)` req_int_terms: `t1 ≡ t2`

Latex:
\mforall{}x:\mBbbR{}.  \mforall{}y:\{y:\mBbbR{}|  x  <  y\}  .    \mexists{}n:\mBbbN{}\msupplus{}.  \mexists{}m:\mBbbZ{}.  ((x  <  (r(m)/r(n)))  \mwedge{}  ((r(m)/r(n))  <  y))

Date html generated: 2020_05_20-AM-11_08_22
Last ObjectModification: 2020_03_20-PM-01_12_09

Theory : reals

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