Nuprl Lemma : not-equal-2
∀x,y:ℤ. uiff(¬(x = y ∈ ℤ);((1 + x) ≤ y) ∨ ((1 + y) ≤ x))
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
or: P ∨ Q,
guard: {T},
sq_type: SQType(T),
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
equal-wf-base,
int_subtype_base,
not_wf,
or_wf,
le_wf,
less-iff-le,
less_than_wf,
less-trichotomy,
subtype_base_sq,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
intEquality,
applyEquality,
hypothesis,
rename,
because_Cache,
addEquality,
natural_numberEquality,
addLevel,
orFunctionality,
productElimination,
independent_isectElimination,
unionElimination,
inlFormation,
inrFormation,
independent_functionElimination,
instantiate,
cumulativity
Latex:
\mforall{}x,y:\mBbbZ{}. uiff(\mneg{}(x = y);((1 + x) \mleq{} y) \mvee{} ((1 + y) \mleq{} x))
Date html generated:
2019_06_20-AM-11_23_05
Last ObjectModification:
2018_08_17-AM-11_33_25
Theory : arithmetic
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