Nuprl Lemma : not-msep
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. x ≡ y supposing ¬x # y
Proof
Definitions occuring in Statement :
msep: x # y,
meq: x ≡ y,
metric: metric(X),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
msep: x # y,
meq: x ≡ y,
metric: metric(X),
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
false: False,
mdist: mdist(d;x;y)
Lemmas referenced :
not-rless,
int-to-real_wf,
mdist_wf,
mdist-nonneg,
req_witness,
msep_wf,
istype-void,
metric_wf,
istype-universe,
req_inversion,
rleq_antisymmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
applyEquality,
setElimination,
rename,
independent_functionElimination,
functionIsType,
universeIsType,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x,y:X]. x \mequiv{} y supposing \mneg{}x \# y
Date html generated:
2019_10_29-AM-11_02_06
Last ObjectModification:
2019_10_02-AM-09_43_10
Theory : reals
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