Nuprl Lemma : geo-between-sep1
∀g:EuclideanPlane. ∀a,b:Point. ∀[x:{x:Point| a_x_b ∧ a ≠ x} ]. a ≠ b
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
and: P ∧ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
euclidean-plane: EuclideanPlane,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
geo-sep_wf,
geo-between_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
set_wf,
geo-between-sep,
sq_stable__geo-sep
Rules used in proof :
because_Cache,
productEquality,
lambdaEquality,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
productElimination,
independent_functionElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b:Point. \mforall{}[x:\{x:Point| a\_x\_b \mwedge{} a \mneq{} x\} ]. a \mneq{} b
Date html generated:
2017_10_02-PM-03_29_01
Last ObjectModification:
2017_08_04-PM-09_07_24
Theory : euclidean!plane!geometry
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