Nuprl Lemma : nequal-le-implies
∀[x,y:ℤ]. ((x + 1) ≤ y) supposing ((x ≤ y) and y ≠ x)
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
nequal: a ≠ b ∈ T ,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
subtract: n - m
Lemmas referenced :
decidable__le,
false_wf,
not-le-2,
not-equal-2,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
minus-one-mul-top,
zero-add,
le-add-cancel2,
less_than'_wf,
le_wf,
nequal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
lemma_by_obid,
dependent_functionElimination,
addEquality,
hypothesisEquality,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[x,y:\mBbbZ{}]. ((x + 1) \mleq{} y) supposing ((x \mleq{} y) and y \mneq{} x)
Date html generated:
2016_05_13-PM-03_31_41
Last ObjectModification:
2015_12_26-AM-09_46_05
Theory : arithmetic
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