Nuprl Lemma : no-nontrivial-decidable-real-prop
∀[A:ℝ ⟶ ℙ]. ((∀x,y:ℝ. ((x = y) ⇒ (A[x] ⇐⇒ A[y]))) ⇒ (∀r:ℝ. (A[r] ∨ (¬A[r]))) ⇒ ((∀x:ℝ. A[x]) ∨ (∀x:ℝ. (¬A[x]))))
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],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
prop: ℙ,
not: ¬A,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
int-to-real_wf,
no-real-separation-corollary,
not_wf,
real_wf,
exists_wf,
req_wf,
all_wf,
or_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
unionElimination,
inlFormation,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_functionElimination,
dependent_pairFormation,
because_Cache,
productElimination,
productEquality,
universeEquality,
voidElimination,
inrFormation,
functionEquality,
cumulativity
Latex:
\mforall{}[A:\mBbbR{} {}\mrightarrow{} \mBbbP{}]
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (A[x] \mLeftarrow{}{}\mRightarrow{} A[y])))
{}\mRightarrow{} (\mforall{}r:\mBbbR{}. (A[r] \mvee{} (\mneg{}A[r])))
{}\mRightarrow{} ((\mforall{}x:\mBbbR{}. A[x]) \mvee{} (\mforall{}x:\mBbbR{}. (\mneg{}A[x]))))
Date html generated:
2017_10_03-AM-10_01_51
Last ObjectModification:
2017_06_30-PM-00_32_14
Theory : reals
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