Nuprl Lemma : not-lt-or
∀g:EuclideanPlane. ∀p,q:Length. (¬¬(p < q ∨ q < p ∨ (p = q ∈ Length)))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length-type: Length,
euclidean-plane: EuclideanPlane,
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
geo-le: p ≤ q,
squash: ↓T,
true: True,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
geo-lt: p < q,
iff: P ⇐⇒ Q,
geo-eq: a ≡ b,
stable: Stable{P},
cand: A c∧ B,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
euclidean-plane: EuclideanPlane,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
geo-length-type: Length,
prop: ℙ,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
member: t ∈ T,
or: P ∨ Q,
false: False,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x]
Lemmas referenced :
geo-length_functionality,
geo-eq_inversion,
or_wf,
geo-length-equality,
squash_wf,
true_wf,
basic-geometry_wf,
geo-length-flip,
subtype_rel_self,
iff_weakening_equal,
geo-between-trivial,
geo-add-length-between,
geo-le_wf,
geo-add-length_wf,
geo-length_wf,
geo-mk-seg_wf,
exists_wf,
equal_wf,
minimal-not-not-excluded-middle,
geo-eq_weakening,
geo-between_functionality,
minimal-double-negation-hyp-elim,
not_wf,
geo-sep_wf,
stable__false,
geo-sep-O-X,
geo-between-same-side2,
geo-length-equiv,
subtype_quotient,
geo-X_wf,
geo-O_wf,
geo-between_wf,
geo-point_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-eq_wf,
false_wf,
euclidean-plane_wf,
geo-length-type_wf,
istype-void,
geo-lt_wf
Rules used in proof :
dependent_set_memberEquality_alt,
inrFormation_alt,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
universeEquality,
natural_numberEquality,
dependent_pairFormation,
productEquality,
lambdaEquality,
inrFormation,
inlFormation,
lambdaFormation,
unionElimination,
functionEquality,
unionEquality,
independent_pairFormation,
lambdaEquality_alt,
setEquality,
dependent_functionElimination,
setIsType,
rename,
setElimination,
applyLambdaEquality,
independent_isectElimination,
instantiate,
applyEquality,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
productIsType,
productElimination,
promote_hyp,
pertypeElimination,
pointwiseFunctionalityForEquality,
inhabitedIsType,
equalityIstype,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
universeIsType,
unionIsType,
functionIsType,
sqequalRule,
voidElimination,
independent_functionElimination,
sqequalHypSubstitution,
hypothesis,
because_Cache,
thin,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane. \mforall{}p,q:Length. (\mneg{}\mneg{}(p < q \mvee{} q < p \mvee{} (p = q)))
Date html generated:
2019_10_29-AM-09_16_11
Last ObjectModification:
2019_10_18-PM-03_23_12
Theory : euclidean!plane!geometry
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