Nuprl Lemma : Vesley-subset-connected
VesleyAxiom
⇒ (∀P:ℝ ⟶ ℙ
((∀x:ℝ. ∀y:{y:ℝ| x = y} . (P[y] ⇒ P[x])) ⇒ dense-in-interval((-∞, ∞);λx.(¬P[x])) ⇒ Connected({x:ℝ| ¬P[x]} )))
Proof
Definitions occuring in Statement :
VesleyAxiom: VesleyAxiom,
connected: Connected(X),
dense-in-interval: dense-in-interval(I;X),
riiint: (-∞, ∞),
req: x = y,
real: ℝ,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
VesleyAxiom: VesleyAxiom,
exists: ∃x:A. B[x],
guard: {T},
uimplies: b supposing a,
false: False
Lemmas referenced :
real-subset-connected,
not_wf,
real_wf,
sq_stable__not,
set_wf,
req_wf,
bool_wf,
dense-in-interval_wf,
riiint_wf,
i-member_wf,
all_wf,
VesleyAxiom_wf,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
isectElimination,
applyEquality,
functionExtensionality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
because_Cache,
setElimination,
rename,
functionEquality,
setEquality,
cumulativity,
universeEquality,
dependent_set_memberEquality,
independent_isectElimination,
voidElimination
Latex:
VesleyAxiom
{}\mRightarrow{} (\mforall{}P:\mBbbR{} {}\mrightarrow{} \mBbbP{}
((\mforall{}x:\mBbbR{}. \mforall{}y:\{y:\mBbbR{}| x = y\} . (P[y] {}\mRightarrow{} P[x]))
{}\mRightarrow{} dense-in-interval((-\minfty{}, \minfty{});\mlambda{}x.(\mneg{}P[x]))
{}\mRightarrow{} Connected(\{x:\mBbbR{}| \mneg{}P[x]\} )))
Date html generated:
2017_10_03-AM-10_15_30
Last ObjectModification:
2017_09_13-PM-04_02_11
Theory : reals
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