Nuprl Lemma : CCC-Sigma02-dns
∀K:Type. (CCCNSet(K) ⇒ (∀P:K ⟶ K ⟶ ℙ. ((∀k,m:K. Dec(P[k;m])) ⇒ (¬¬(∃k:K. ∀m:K. P[k;m])) ⇒ (∃k:K. ∀m:K. P[k;m]))))
Proof
Definitions occuring in Statement :
ccc-nset: CCCNSet(K),
decidable: Dec(P),
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
false: False,
not: ¬A,
or: P ∨ Q,
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
istype-universe,
ccc-nset_wf,
decidable_wf,
subtype_rel_self,
istype-void,
CCC-omni-2
Rules used in proof :
productElimination,
universeEquality,
isectElimination,
instantiate,
applyEquality,
Error :universeIsType,
Error :productIsType,
because_Cache,
voidElimination,
Error :inhabitedIsType,
Error :functionIsType,
sqequalRule,
unionElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}K:Type
(CCCNSet(K)
{}\mRightarrow{} (\mforall{}P:K {}\mrightarrow{} K {}\mrightarrow{} \mBbbP{}. ((\mforall{}k,m:K. Dec(P[k;m])) {}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}k:K. \mforall{}m:K. P[k;m])) {}\mRightarrow{} (\mexists{}k:K. \mforall{}m:K. P[k;m]))))
Date html generated:
2019_06_20-PM-03_02_59
Last ObjectModification:
2019_06_14-AM-10_04_54
Theory : continuity
Home
Index