Nuprl Lemma : not-all-finite
∀[T:Type]. ∀[P:T ⟶ ℙ]. ((∀x:T. Dec(P[x])) ⇒ (∃k:ℕ. T ~ ℕk) ⇒ (¬(∀x:T. P[x]) ⇐⇒ ∃x:T. (¬P[x])))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
true: True,
inject: Inj(A;B;f),
pi1: fst(t),
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
uimplies: b supposing a,
int_seg: {i..j-},
or: P ∨ Q,
decidable: Dec(P),
so_lambda: λ2x.t[x],
surject: Surj(A;B;f),
biject: Bij(A;B;f),
equipollent: A ~ B,
guard: {T},
nat: ℕ,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
member: t ∈ T,
all: ∀x:A. B[x],
not: ¬A,
exists: ∃x:A. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
iff_weakening_equal,
true_wf,
squash_wf,
equal_wf,
int_subtype_base,
istype-int,
lelt_wf,
set_subtype_base,
not-all-int_seg,
decidable__all_int_seg,
istype-universe,
decidable_wf,
int_seg_wf,
equipollent_wf,
istype-nat,
istype-void,
subtype_rel_self
Rules used in proof :
baseClosed,
imageMemberEquality,
hyp_replacement,
equalityTransitivity,
Error :inhabitedIsType,
functionExtensionality,
Error :dependent_pairFormation_alt,
equalitySymmetry,
sqequalBase,
imageElimination,
independent_isectElimination,
intEquality,
Error :equalityIstype,
unionElimination,
Error :lambdaEquality_alt,
promote_hyp,
dependent_functionElimination,
rename,
setElimination,
natural_numberEquality,
Error :productIsType,
because_Cache,
voidElimination,
independent_functionElimination,
universeEquality,
isectElimination,
extract_by_obid,
introduction,
instantiate,
hypothesis,
applyEquality,
cut,
hypothesisEquality,
Error :universeIsType,
Error :functionIsType,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:T. Dec(P[x])) {}\mRightarrow{} (\mexists{}k:\mBbbN{}. T \msim{} \mBbbN{}k) {}\mRightarrow{} (\mneg{}(\mforall{}x:T. P[x]) \mLeftarrow{}{}\mRightarrow{} \mexists{}x:T. (\mneg{}P[x])))
Date html generated:
2019_06_20-PM-02_19_39
Last ObjectModification:
2019_06_06-PM-00_37_22
Theory : equipollence!!cardinality!
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