Nuprl Lemma : not-less-implies-equal
∀x,y:ℤ. (x = y ∈ ℤ) supposing ((¬x < y) and (¬y < x))
Proof
Definitions occuring in Statement :
less_than: a < b,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
decidable__int_equal,
false_wf,
not-equal-2,
not-lt-2,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
add-associates,
not_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
addEquality,
natural_numberEquality,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x,y:\mBbbZ{}. (x = y) supposing ((\mneg{}x < y) and (\mneg{}y < x))
Date html generated:
2016_05_13-PM-03_32_27
Last ObjectModification:
2015_12_26-AM-09_45_20
Theory : arithmetic
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