Nuprl Lemma : le-iff-less-or-equal
∀x,y:ℤ. uiff(x ≤ y;x < y ∨ (x = y ∈ ℤ))
Proof
Definitions occuring in Statement :
less_than: a < b,
uiff: uiff(P;Q),
le: A ≤ B,
all: ∀x:A. B[x],
or: P ∨ Q,
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,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
guard: {T},
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
sq_type: SQType(T),
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
less_than'_wf,
le_wf,
or_wf,
less_than_wf,
equal-wf-base,
int_subtype_base,
less-trichotomy,
less_than_irreflexivity,
less_than_transitivity,
subtype_base_sq,
top_wf,
less_than_anti-reflexive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
hypothesis,
axiomEquality,
rename,
intEquality,
applyEquality,
unionElimination,
inlFormation,
inrFormation,
imageElimination,
independent_functionElimination,
because_Cache,
imageMemberEquality,
baseClosed,
independent_isectElimination,
instantiate,
cumulativity,
isect_memberEquality,
voidEquality,
lessCases,
axiomSqEquality,
natural_numberEquality
Latex:
\mforall{}x,y:\mBbbZ{}. uiff(x \mleq{} y;x < y \mvee{} (x = y))
Date html generated:
2019_06_20-AM-11_22_46
Last ObjectModification:
2018_09_10-PM-01_12_31
Theory : arithmetic
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