Nuprl Lemma : geo-sep-exists
∀e:EuclideanPlane. ∀A:Point. ∃A':Point. A ≠ A'
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x]
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
or: P ∨ Q
Lemmas referenced :
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
geo-sep_wf,
geo-X_wf,
geo-sep-O-X,
geo-O_wf,
geo-sep-or,
geo-sep-sym
Rules used in proof :
independent_isectElimination,
instantiate,
sqequalRule,
applyEquality,
isectElimination,
dependent_set_memberEquality,
because_Cache,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_pairFormation,
unionElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}A:Point. \mexists{}A':Point. A \mneq{} A'
Date html generated:
2017_10_02-PM-03_28_33
Last ObjectModification:
2017_08_04-PM-09_03_15
Theory : euclidean!plane!geometry
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