Nuprl Lemma : not-col-distincts
∀e:BasicGeometry. ∀A,B,C:Point. ((¬Colinear(A;B;C)) ⇒ (¬¬(A ≠ B ∧ B ≠ C ∧ A ≠ C)))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
geo-eq: a ≡ b,
or: P ∨ Q,
stable: Stable{P},
cand: A c∧ B,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
minimal-not-not-excluded-middle,
geo-colinear_functionality,
geo-eq_weakening,
geo-sep_functionality,
minimal-double-negation-hyp-elim,
geo-colinear-same,
or_wf,
false_wf,
stable__false,
geo-point_wf,
geo-colinear_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
not_wf
Rules used in proof :
impliesLevelFunctionality,
promote_hyp,
levelHypothesis,
andLevelFunctionality,
dependent_functionElimination,
impliesFunctionality,
addLevel,
unionElimination,
productElimination,
independent_pairFormation,
functionEquality,
rename,
setElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
productEquality,
isectElimination,
extract_by_obid,
introduction,
voidElimination,
independent_functionElimination,
sqequalHypSubstitution,
hypothesis,
because_Cache,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,C:Point. ((\mneg{}Colinear(A;B;C)) {}\mRightarrow{} (\mneg{}\mneg{}(A \mneq{} B \mwedge{} B \mneq{} C \mwedge{} A \mneq{} C)))
Date html generated:
2017_10_02-PM-06_32_12
Last ObjectModification:
2017_08_05-PM-04_43_01
Theory : euclidean!plane!geometry
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