Nuprl Lemma : geo-nontrivial_wf
∀[g:EuclideanPlaneStructure]. (geo-nontrivial(g) ∈ ∃a:Point. (∃b:{Point| a ≠ b}))
Proof
Definitions occuring in Statement :
geo-nontrivial: geo-nontrivial(e),
euclidean-plane-structure: EuclideanPlaneStructure,
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
implies: P ⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
or: P ∨ Q,
all: ∀x:A. B[x],
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
subtype_rel: A ⊆r B,
record-select: r.x,
record+: record+,
euclidean-plane-structure: EuclideanPlaneStructure,
geo-nontrivial: geo-nontrivial(e),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
euclidean-plane-structure_wf,
geo-gt_wf,
geo-ge_wf,
geo-colinear_wf,
sq_exists_wf,
exists_wf,
or_wf,
geo-left_wf,
geo-sep_wf,
sq_stable_wf,
all_wf,
geo-congruent_wf,
geo-between_wf,
stable_wf,
geo-point_wf,
uall_wf,
subtype_rel_self
Rules used in proof :
axiomEquality,
equalitySymmetry,
equalityTransitivity,
functionEquality,
productElimination,
productEquality,
lambdaFormation,
because_Cache,
rename,
setElimination,
setEquality,
hypothesisEquality,
lambdaEquality,
isectElimination,
extract_by_obid,
tokenEquality,
applyEquality,
hypothesis,
thin,
dependentIntersectionEqElimination,
dependentIntersectionElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[g:EuclideanPlaneStructure]. (geo-nontrivial(g) \mmember{} \mexists{}a:Point. (\mexists{}b:\{Point| a \mneq{} b\}))
Date html generated:
2017_10_02-PM-03_28_20
Last ObjectModification:
2017_08_04-PM-08_49_18
Theory : euclidean!plane!geometry
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