Nuprl Lemma : Unique
∀g:BasicProjectivePlane. ∀[l,m:Line]. ∀[p,q:Point]. ¬((¬p ≡ q) ∧ (¬l ≡ m)) supposing p I l ∧ q I l ∧ p I m ∧ q I m
Proof
Definitions occuring in Statement :
basic-projective-plane: BasicProjectivePlane,
pgeo-leq: a ≡ b,
pgeo-peq: a ≡ b,
pgeo-incident: a I b,
pgeo-line: Line,
pgeo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
basic-projective-plane-axioms,
not_wf,
pgeo-peq_wf,
pgeo-leq_wf,
pgeo-incident_wf,
projective-plane-structure_subtype,
basic-projective-plane-subtype,
subtype_rel_transitivity,
basic-projective-plane_wf,
projective-plane-structure_wf,
pgeo-primitives_wf,
pgeo-point_wf,
pgeo-line_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
productElimination,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
voidElimination,
because_Cache,
productEquality,
isectElimination,
applyEquality,
sqequalRule,
lambdaEquality,
instantiate,
independent_isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}g:BasicProjectivePlane
\mforall{}[l,m:Line]. \mforall{}[p,q:Point]. \mneg{}((\mneg{}p \mequiv{} q) \mwedge{} (\mneg{}l \mequiv{} m)) supposing p I l \mwedge{} q I l \mwedge{} p I m \mwedge{} q I m
Date html generated:
2018_05_22-PM-00_34_43
Last ObjectModification:
2017_11_10-PM-03_46_45
Theory : euclidean!plane!geometry
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