Nuprl Lemma : not-not-finite-xmiddle-1
∀[T:Type]. ((∃n:ℕ. ∃f:ℕn ⟶ T. Surj(ℕn;T;f)) ⇒ (∀P:T ⟶ ℙ. (¬¬(∀i:T. (P[i] ∨ (¬P[i]))))))
Proof
Definitions occuring in Statement :
surject: Surj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
exists: ∃x:A. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
surject: Surj(A;B;f),
or: P ∨ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not-not-all-int_seg-xmiddle,
int_seg_wf,
istype-void,
subtype_rel_self,
istype-nat,
surject_wf,
istype-universe,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
thin,
sqequalHypSubstitution,
productElimination,
extract_by_obid,
dependent_functionElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
universeIsType,
isectElimination,
independent_functionElimination,
voidElimination,
functionIsType,
unionIsType,
instantiate,
universeEquality,
productIsType,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
inlFormation_alt,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inrFormation_alt
Latex:
\mforall{}[T:Type]. ((\mexists{}n:\mBbbN{}. \mexists{}f:\mBbbN{}n {}\mrightarrow{} T. Surj(\mBbbN{}n;T;f)) {}\mRightarrow{} (\mforall{}P:T {}\mrightarrow{} \mBbbP{}. (\mneg{}\mneg{}(\mforall{}i:T. (P[i] \mvee{} (\mneg{}P[i]))))))
Date html generated:
2020_05_19-PM-10_00_45
Last ObjectModification:
2019_10_24-AM-10_43_14
Theory : equipollence!!cardinality!
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