Nuprl Lemma : msep-or
∀[X:Type]. ∀d:metric(X). ∀x,y:X. (x # y ⇒ (∀z:X. (x # z ∨ z # y)))
Proof
Definitions occuring in Statement :
msep: x # y,
metric: metric(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
msep: x # y,
guard: {T},
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
mdist-triangle-inequality,
rless_transitivity1,
int-to-real_wf,
mdist_wf,
radd_wf,
msep_wf,
metric_wf,
istype-universe,
radd-positive-implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
independent_functionElimination,
independent_isectElimination,
inhabitedIsType,
universeIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}d:metric(X). \mforall{}x,y:X. (x \# y {}\mRightarrow{} (\mforall{}z:X. (x \# z \mvee{} z \# y)))
Date html generated:
2019_10_29-AM-11_01_44
Last ObjectModification:
2019_10_02-AM-09_42_52
Theory : reals
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