Nuprl Lemma : not-not-inner-pasch
∀e:EuclideanPlane. ∀a,b,c:Point. ∀p:{p:Point| a_p_c} . ∀q:{q:Point| b_q_c} . (¬¬(∃x:Point. (p_x_b ∧ q_x_a)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
stable: Stable{P},
uimplies: b supposing a,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
cand: A c∧ B,
rev_implies: P ⇐ Q
Lemmas referenced :
not_wf,
exists_wf,
eu-point_wf,
eu-between-eq_wf,
set_wf,
euclidean-plane_wf,
eu-colinear-cases,
equal_wf,
eu-between_wf,
eu-colinear_wf,
dneg_elim_a,
all_wf,
stable_wf,
eu-between-eq-same,
eu-between-eq-symmetry,
eu-between-eq-trivial-left,
eu-between-eq-trivial-right,
eu-between-implies-between-eq,
eu-between-eq-exchange3,
eu-between-eq-inner-trans,
eu-between-eq-exchange4,
eu-between-eq-def,
eu-inner-pasch-property,
eu-inner-pasch_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
setElimination,
rename,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
hypothesisEquality,
dependent_functionElimination,
isect_memberFormation,
equalityEquality,
addLevel,
impliesFunctionality,
productElimination,
levelHypothesis,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
instantiate,
universeEquality,
functionEquality,
applyEquality,
cumulativity,
independent_isectElimination,
dependent_pairFormation,
independent_pairFormation,
promote_hyp,
dependent_set_memberEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. \mforall{}p:\{p:Point| a\_p\_c\} . \mforall{}q:\{q:Point| b\_q\_c\} .
(\mneg{}\mneg{}(\mexists{}x:Point. (p\_x\_b \mwedge{} q\_x\_a)))
Date html generated:
2016_10_26-AM-07_41_25
Last ObjectModification:
2016_07_12-AM-08_08_24
Theory : euclidean!geometry
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