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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