Nuprl Lemma : geo-sep-iff-or-lt
∀e:BasicGeometry. ∀x,y:{p:Point| O_X_p} . (x ≠ y ⇐⇒ x < y ∨ y < x)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
basic-geometry: BasicGeometry,
geo-X: X,
geo-O: O,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
basic-geometry-: BasicGeometry-,
geo-lt: p < q,
exists: ∃x:A. B[x],
geo-le: p ≤ q,
true: True
Lemmas referenced :
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lt_wf,
subtype-geo-length-type,
geo-point_wf,
geo-between_wf,
geo-O_wf,
geo-X_wf,
geo-between-same-side-or,
geo-sep-O-X,
sq_stable__geo-between,
geo-between-symmetry,
geo-between-inner-trans,
geo-between-exchange3,
subtype_rel_self,
basic-geometry-_wf,
geo-add-length-between,
geo-le_wf,
geo-add-length_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-between-trivial,
geo-length-equality,
equal_wf,
squash_wf,
true_wf,
istype-universe,
geo-length-type_wf,
iff_weakening_equal,
geo-sep-sym,
geo-lt-iff-strict-between-points
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
setElimination,
rename,
unionIsType,
because_Cache,
inhabitedIsType,
setIsType,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
inlFormation_alt,
inrFormation_alt,
dependent_pairFormation_alt,
productIsType,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
lambdaEquality_alt,
universeEquality,
natural_numberEquality,
productElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}x,y:\{p:Point| O\_X\_p\} . (x \mneq{} y \mLeftarrow{}{}\mRightarrow{} x < y \mvee{} y < x)
Date html generated:
2019_10_16-PM-01_37_55
Last ObjectModification:
2019_02_18-PM-05_08_39
Theory : euclidean!plane!geometry
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