Nuprl Lemma : geo-bet-or
∀g:EuclideanPlane. ∀a,b,c,d:Point. (((a_b_c ∧ a_b_d) ∧ a ≠ b) ⇒ (¬¬(b_c_d ∨ b_d_c)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
and: P ∧ Q,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
cand: A c∧ B,
or: P ∨ Q,
basic-geometry-: BasicGeometry-,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
geo-between-same-side,
geo-between-symmetry,
geo-between-inner-trans,
geo-between-exchange3,
geo-between_wf,
not_wf,
or_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
productElimination,
sqequalRule,
hypothesisEquality,
isectElimination,
independent_isectElimination,
hypothesis,
independent_functionElimination,
inlFormation,
because_Cache,
applyEquality,
voidElimination,
independent_pairFormation,
inrFormation,
instantiate,
productEquality
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (((a\_b\_c \mwedge{} a\_b\_d) \mwedge{} a \mneq{} b) {}\mRightarrow{} (\mneg{}\mneg{}(b\_c\_d \mvee{} b\_d\_c)))
Date html generated:
2019_10_16-PM-02_50_02
Last ObjectModification:
2018_09_20-AM-10_02_51
Theory : euclidean!plane!geometry
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