Nuprl Lemma : not-equal-implies-less
∀[T:Type]. ∀x,y:T. ((¬(x = y ∈ T)) ⇒ (x < y ∨ y < x)) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
less_than: a < b,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
equal_wf,
not_wf,
subtype_rel_wf,
decidable__lt,
less_than_wf,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
or_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
sqequalHypSubstitution,
independent_functionElimination,
instantiate,
extract_by_obid,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
voidElimination,
applyEquality,
because_Cache,
universeEquality,
unionElimination,
inlFormation,
independent_pairFormation,
productElimination,
inrFormation,
addEquality,
natural_numberEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
addLevel,
orFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}x,y:T. ((\mneg{}(x = y)) {}\mRightarrow{} (x < y \mvee{} y < x)) supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2017_04_14-AM-07_16_36
Last ObjectModification:
2017_02_27-PM-02_51_34
Theory : arithmetic
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