Nuprl Lemma : no-real-separation
∀[A,B:ℝ ⟶ ℙ]. (¬real-separation(x.A[x];y.B[y]))
Proof
Definitions occuring in Statement :
real-separation: real-separation(x.A[x];y.B[y]),
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
not: ¬A,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
real-separation: real-separation(x.A[x];y.B[y]),
and: P ∧ Q,
or: P ∨ Q,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
true: True,
real-disjoint: real-disjoint(x.A[x];y.B[y]),
cand: A c∧ B,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
isr: isr(x),
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
bnot: ¬bb
Lemmas referenced :
real-separation_wf,
real_wf,
false_wf,
or_wf,
exists_wf,
req_wf,
assert_wf,
isl_wf,
isr_wf,
true_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
equal_wf,
extensional-real-to-bool-constant,
bool_wf,
eqtt_to_assert,
btrue_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
not_wf,
req_inversion,
equal-wf-base,
btrue_neq_bfalse,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
productElimination,
sqequalRule,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
dependent_functionElimination,
functionEquality,
cumulativity,
universeEquality,
isect_memberEquality,
productEquality,
unionElimination,
natural_numberEquality,
independent_pairFormation,
unionEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
baseClosed
Latex:
\mforall{}[A,B:\mBbbR{} {}\mrightarrow{} \mBbbP{}]. (\mneg{}real-separation(x.A[x];y.B[y]))
Date html generated:
2017_10_03-AM-10_01_23
Last ObjectModification:
2017_06_30-AM-11_29_51
Theory : reals
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