Nuprl Lemma : midpoint-sep
∀e:BasicGeometry. ∀A,B,M:Point.  (A ≠ B 
⇒ A=M=B 
⇒ {A ≠ M ∧ B ≠ M})
Proof
Definitions occuring in Statement : 
geo-midpoint: a=m=b
, 
basic-geometry: BasicGeometry
, 
geo-sep: a ≠ b
, 
geo-point: Point
, 
guard: {T}
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
Definitions unfolded in proof : 
uimplies: b supposing a
, 
or: P ∨ Q
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
basic-geometry: BasicGeometry
, 
member: t ∈ T
, 
cand: A c∧ B
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
guard: {T}
, 
geo-midpoint: a=m=b
Lemmas referenced : 
geo-point_wf, 
geo-congruent_wf, 
Error :basic-geo-primitives_wf, 
Error :basic-geo-structure_wf, 
basic-geometry_wf, 
subtype_rel_transitivity, 
basic-geometry-subtype, 
geo-between_wf, 
geo-congruent-symmetry, 
geo-sep-sym, 
geo-congruent-sep, 
geo-sep_wf, 
geo-sep-or
Rules used in proof : 
instantiate, 
productEquality, 
independent_isectElimination, 
independent_pairFormation, 
independent_functionElimination, 
unionElimination, 
because_Cache, 
applyEquality, 
isectElimination, 
dependent_set_memberEquality, 
hypothesis, 
hypothesisEquality, 
rename, 
setElimination, 
dependent_functionElimination, 
extract_by_obid, 
introduction, 
cut, 
thin, 
productElimination, 
sqequalHypSubstitution, 
lambdaFormation, 
computationStep, 
sqequalTransitivity, 
sqequalReflexivity, 
sqequalRule, 
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry.  \mforall{}A,B,M:Point.    (A  \mneq{}  B  {}\mRightarrow{}  A=M=B  {}\mRightarrow{}  \{A  \mneq{}  M  \mwedge{}  B  \mneq{}  M\})
Date html generated:
2017_10_02-PM-06_34_12
Last ObjectModification:
2017_08_05-PM-04_44_16
Theory : euclidean!plane!geometry
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