Nuprl Lemma : msep-not-meq
∀[X:Type]. ∀d:metric(X). ∀x,y:X. (x # y ⇒ (¬x ≡ y))
Proof
Definitions occuring in Statement :
msep: x # y,
meq: x ≡ y,
metric: metric(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
msep: x # y,
meq: x ≡ y,
mdist: mdist(d;x;y),
guard: {T},
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
rless_transitivity1,
int-to-real_wf,
mdist_wf,
rleq_weakening,
rless_irreflexivity,
meq_wf,
msep_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
thin,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
hypothesisEquality,
because_Cache,
independent_functionElimination,
independent_isectElimination,
voidElimination,
universeIsType,
inhabitedIsType,
sqequalRule,
lambdaEquality_alt,
functionIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}d:metric(X). \mforall{}x,y:X. (x \# y {}\mRightarrow{} (\mneg{}x \mequiv{} y))
Date html generated:
2019_10_29-AM-11_02_28
Last ObjectModification:
2019_10_02-AM-09_43_27
Theory : reals
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