Nuprl Lemma : pgeo-plsep-cycle
∀g:ProjectivePlane. ∀a,b,c:Point. ∀s:a ≠ b. ∀s1:b ≠ c. ∀s2:a ≠ c. (a ≠ b ∨ c ⇒ {b ≠ a ∨ c ∧ c ≠ a ∨ b})
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-join: p ∨ q,
pgeo-psep: a ≠ b,
pgeo-plsep: a ≠ b,
pgeo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
false: False,
cand: A c∧ B
Lemmas referenced :
PL-sep-or,
pgeo-join_wf,
projective-plane-structure-complete_subtype,
projective-plane-subtype,
subtype_rel_transitivity,
projective-plane_wf,
projective-plane-structure-complete_wf,
projective-plane-structure_wf,
pgeo-line_wf,
pgeo-incident_wf,
psep-join-implies-false,
pgeo-plsep_wf,
projective-plane-structure_subtype,
pgeo-primitives_wf,
pgeo-psep_wf,
pgeo-point_wf,
pgeo-lsep-or,
projective-plane-subtype-basic,
pgeo-lsep-implies-plsep_dual,
incident-join-first,
pgeo-lsep-implies-plsep,
incident-join-second,
pgeo-meet-implies-psep2,
pgeo-psep-sym,
pgeo-lsep-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
independent_functionElimination,
unionElimination,
voidElimination,
independent_pairFormation,
dependent_set_memberEquality,
productElimination
Latex:
\mforall{}g:ProjectivePlane. \mforall{}a,b,c:Point. \mforall{}s:a \mneq{} b. \mforall{}s1:b \mneq{} c. \mforall{}s2:a \mneq{} c.
(a \mneq{} b \mvee{} c {}\mRightarrow{} \{b \mneq{} a \mvee{} c \mwedge{} c \mneq{} a \mvee{} b\})
Date html generated:
2018_05_22-PM-00_50_46
Last ObjectModification:
2017_12_05-AM-11_04_43
Theory : euclidean!plane!geometry
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