Nuprl Lemma : Vesley-connected-rationals
VesleyAxiom ⇒ Connected({x:ℝ| ¬¬(∃a:ℤ. ∃b:ℤ-o. (x = (r(a)/r(b))))} )
Proof
Definitions occuring in Statement :
VesleyAxiom: VesleyAxiom,
connected: Connected(X),
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
real: ℝ,
int_nzero: ℤ-o,
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
int: ℤ
Definitions unfolded in proof :
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
sq_stable: SqStable(P),
guard: {T},
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
dense-in-interval: dense-in-interval(I;X),
nat_plus: ℕ+,
rneq: x ≠ y,
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
cand: A c∧ B
Latex:
VesleyAxiom {}\mRightarrow{} Connected(\{x:\mBbbR{}| \mneg{}\mneg{}(\mexists{}a:\mBbbZ{}. \mexists{}b:\mBbbZ{}\msupminus{}\msupzero{}. (x = (r(a)/r(b))))\} )
Date html generated:
2020_05_20-PM-00_07_26
Last ObjectModification:
2020_01_06-PM-00_21_57
Theory : reals
Home
Index