### Nuprl Lemma : eu-eq-x-implies-eq

`∀e:EuclideanPlane. ∀a,b:Point.  ((X = |ab| ∈ {p:Point| O_X_p} ) `` (a = b ∈ Point))`

Proof

Definitions occuring in Statement :  eu-length: `|s|` eu-mk-seg: `ab` euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-X: `X` eu-O: `O` eu-point: `Point` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` eu-length: `|s|` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` top: `Top` squash: `↓T` true: `True`
Lemmas referenced :  eu-segment_wf and_wf euclidean-structure_wf true_wf squash_wf eu-congruent-iff-length eu_seg1_mk_seg_lemma eu_seg2_mk_seg_lemma eu-congruence-identity-sym eu-not-colinear-OXY eu-seg2_wf eu-seg1_wf eu-extend-equal-iff-congruent iff_weakening_equal eu-length-null-segment euclidean-plane_wf eu-mk-seg_wf eu-length_wf eu-between-eq-trivial-right eu-X_wf eu-O_wf eu-between-eq_wf eu-point_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis lemma_by_obid sqequalHypSubstitution isectElimination thin setEquality setElimination rename hypothesisEquality because_Cache dependent_functionElimination dependent_set_memberEquality equalityEquality applyEquality lambdaEquality sqequalRule equalityTransitivity equalitySymmetry independent_isectElimination productElimination independent_functionElimination isect_memberEquality voidElimination voidEquality imageElimination natural_numberEquality imageMemberEquality baseClosed independent_pairFormation

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b:Point.    ((X  =  |ab|)  {}\mRightarrow{}  (a  =  b))

Date html generated: 2016_05_18-AM-06_44_00
Last ObjectModification: 2016_01_16-PM-10_28_53

Theory : euclidean!geometry

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