Nuprl Lemma : no-real-separation-corollary
∀[A,B:ℝ ⟶ ℙ]. ((∃x:ℝ. A[x]) ⇒ (∃y:ℝ. B[y]) ⇒ (∀r:ℝ. (A[r] ∨ B[r])) ⇒ (¬¬(∃x,y:ℝ. ((x = y) ∧ A[x] ∧ B[y]))))
Proof
Definitions occuring in Statement :
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
false: False,
real-separation: real-separation(x.A[x];y.B[y]),
and: P ∧ Q,
real-disjoint: real-disjoint(x.A[x];y.B[y]),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
cand: A c∧ B,
or: P ∨ Q
Lemmas referenced :
no-real-separation,
req_wf,
real_wf,
exists_wf,
not_wf,
all_wf,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
productElimination,
hypothesis,
dependent_pairFormation,
because_Cache,
productEquality,
applyEquality,
functionExtensionality,
sqequalRule,
lambdaEquality,
voidElimination,
dependent_functionElimination,
functionEquality,
cumulativity,
universeEquality,
isect_memberEquality
Latex:
\mforall{}[A,B:\mBbbR{} {}\mrightarrow{} \mBbbP{}].
((\mexists{}x:\mBbbR{}. A[x]) {}\mRightarrow{} (\mexists{}y:\mBbbR{}. B[y]) {}\mRightarrow{} (\mforall{}r:\mBbbR{}. (A[r] \mvee{} B[r])) {}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}x,y:\mBbbR{}. ((x = y) \mwedge{} A[x] \mwedge{} B[y]))))
Date html generated:
2017_10_03-AM-10_01_36
Last ObjectModification:
2017_06_30-PM-00_24_04
Theory : reals
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