Nuprl Lemma : rneq-int
∀n,m:ℤ. (r(n) ≠ r(m) ⇐⇒ ¬(n = m ∈ ℤ))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
int-to-real: r(n),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
rneq: x ≠ y,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
iff_wf,
all_wf,
int-to-real_wf,
rless_wf,
rless-int,
not_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
decidable__or,
less_than_wf,
or_wf,
equal_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_formula_prop_or_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
intformor_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
lambdaFormation,
independent_pairFormation,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesis,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
hypothesisEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
because_Cache,
unionElimination,
addLevel,
allFunctionality,
productElimination,
impliesFunctionality,
orFunctionality,
orLevelFunctionality
Latex:
\mforall{}n,m:\mBbbZ{}. (r(n) \mneq{} r(m) \mLeftarrow{}{}\mRightarrow{} \mneg{}(n = m))
Date html generated:
2016_05_18-AM-07_10_27
Last ObjectModification:
2016_01_17-AM-01_52_03
Theory : reals
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