Nuprl Lemma : segment-density-strict
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . (∃x:{Point| a-x-b})
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
Euclid-Prop10,
geo-strict-between_wf,
set_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
instantiate,
independent_isectElimination,
lambdaEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . (\mexists{}x:\{Point| a-x-b\})
Date html generated:
2017_10_02-PM-06_55_25
Last ObjectModification:
2017_08_14-PM-03_40_31
Theory : euclidean!plane!geometry
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