Nuprl Lemma : least-upper-bound-unique
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
∀[a,b,c,d:T].
(c = d ∈ T) supposing (least-upper-bound(T;x,y.R[x;y];a;b;c) and least-upper-bound(T;x,y.R[x;y];a;b;d))
supposing Order(T;x,y.R[x;y])
Proof
Definitions occuring in Statement :
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c)
,
order: Order(T;x,y.R[x; y])
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
prop: ℙ
,
so_apply: x[s1;s2]
,
function: x:A ⟶ B[x]
,
universe: Type
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
prop: ℙ
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2]
,
guard: {T}
,
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c)
,
and: P ∧ Q
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
order: Order(T;x,y.R[x; y])
,
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
least-upper-bound_wf,
order_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}[a,b,c,d:T].
(c = d) supposing
(least-upper-bound(T;x,y.R[x;y];a;b;c) and
least-upper-bound(T;x,y.R[x;y];a;b;d))
supposing Order(T;x,y.R[x;y])
Date html generated:
2016_05_13-PM-04_18_10
Last ObjectModification:
2015_12_26-AM-11_27_29
Theory : rel_1
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