Nuprl Lemma : unbounded-ccc-nset-decidable
∀K:Type. (CCCNSet(K) ⇒ (∀B:ℕ. ∃k:K. B < k) ⇒ (∀l:ℕ. ((l ∈ K) ∨ (¬(l ∈ K)))))
Proof
Definitions occuring in Statement :
ccc-nset: CCCNSet(K),
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
false: False,
so_apply: x[s],
so_lambda: λ2x.t[x],
not: ¬A,
lelt: i ≤ j < k,
int_seg: {i..j-},
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
nat: ℕ,
uall: ∀[x:A]. B[x],
weakly-decidable-nset: WD(K),
exists: ∃x:A. B[x],
and: P ∧ Q,
ccc-nset: CCCNSet(K),
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
istype-le,
decidable__le,
istype-universe,
ccc-nset_wf,
nat_wf,
subtype_rel_transitivity,
istype-less_than,
istype-nat,
ccc-nset-minimum,
ccc-nset-weakly-decidable
Rules used in proof :
voidElimination,
equalityTransitivity,
equalitySymmetry,
sqequalBase,
natural_numberEquality,
Error :equalityIstype,
Error :inrFormation_alt,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
unionElimination,
universeEquality,
instantiate,
independent_isectElimination,
intEquality,
Error :inhabitedIsType,
Error :lambdaEquality_alt,
applyEquality,
rename,
setElimination,
isectElimination,
Error :universeIsType,
Error :productIsType,
Error :functionIsType,
sqequalRule,
because_Cache,
productElimination,
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{}B:\mBbbN{}. \mexists{}k:K. B < k) {}\mRightarrow{} (\mforall{}l:\mBbbN{}. ((l \mmember{} K) \mvee{} (\mneg{}(l \mmember{} K)))))
Date html generated:
2019_06_20-PM-03_02_36
Last ObjectModification:
2019_06_13-PM-04_29_07
Theory : continuity
Home
Index