### Nuprl Lemma : eu-nontrivial_wf

`∀[e:EuclideanStructure]`
`  (eu-nontrivial(e) ∈ {triple:Point × Point × Point| let a,b,c = triple in (¬(a = b ∈ Point)) ∧ (¬Colinear(a;b;c))} )`

Proof

Definitions occuring in Statement :  eu-nontrivial: `eu-nontrivial(e)` eu-colinear: `Colinear(a;b;c)` eu-point: `Point` euclidean-structure: `EuclideanStructure` spreadn: spread3 uall: `∀[x:A]. B[x]` not: `¬A` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` product: `x:A × B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` eu-nontrivial: `eu-nontrivial(e)` eu-colinear: `Colinear(a;b;c)` eu-point: `Point` euclidean-structure: `EuclideanStructure` record+: record+ record-select: `r.x` subtype_rel: `A ⊆r B` eq_atom: `x =a y` ifthenelse: `if b then t else f fi ` btrue: `tt` guard: `{T}` prop: `ℙ` spreadn: spread3 and: `P ∧ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` uimplies: `b supposing a` all: `∀x:A. B[x]`
Lemmas referenced :  subtype_rel_self not_wf equal_wf uall_wf iff_wf and_wf isect_wf euclidean-structure_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution dependentIntersectionElimination dependentIntersectionEqElimination thin hypothesis applyEquality tokenEquality instantiate lemma_by_obid isectElimination universeEquality functionEquality equalityTransitivity equalitySymmetry lambdaEquality cumulativity hypothesisEquality because_Cache setEquality productEquality productElimination setElimination rename lambdaFormation axiomEquality

Latex:
\mforall{}[e:EuclideanStructure]
(eu-nontrivial(e)  \mmember{}  \{triple:Point  \mtimes{}  Point  \mtimes{}  Point|
let  a,b,c  =  triple  in
(\mneg{}(a  =  b))  \mwedge{}  (\mneg{}Colinear(a;b;c))\}  )

Date html generated: 2016_05_18-AM-06_32_31
Last ObjectModification: 2015_12_28-AM-09_28_40

Theory : euclidean!geometry

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