Nuprl Lemma : not-not-finite-exists-or-all
∀[T:Type]. (finite(T) ⇒ (∀P:T ⟶ ℙ. (¬¬((∃i:T. P[i]) ∨ (∀i:T. (¬P[i]))))))
Proof
Definitions occuring in Statement :
finite: finite(T),
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],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
finite: finite(T),
exists: ∃x:A. B[x],
false: False,
or: P ∨ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
equipollent: A ~ B,
decidable: Dec(P),
nat: ℕ,
so_lambda: λ2x.t[x],
biject: Bij(A;B;f),
and: P ∧ Q,
surject: Surj(A;B;f),
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not-not-finite-xmiddle,
istype-void,
subtype_rel_self,
finite_wf,
istype-universe,
equipollent_inversion,
int_seg_wf,
decidable__exists_int_seg,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
independent_functionElimination,
dependent_functionElimination,
productElimination,
voidElimination,
sqequalRule,
functionIsType,
universeIsType,
unionIsType,
applyEquality,
because_Cache,
productIsType,
instantiate,
universeEquality,
natural_numberEquality,
setElimination,
rename,
lambdaEquality_alt,
unionElimination,
inlFormation_alt,
inrFormation_alt,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination
Latex:
\mforall{}[T:Type]. (finite(T) {}\mRightarrow{} (\mforall{}P:T {}\mrightarrow{} \mBbbP{}. (\mneg{}\mneg{}((\mexists{}i:T. P[i]) \mvee{} (\mforall{}i:T. (\mneg{}P[i]))))))
Date html generated:
2020_05_19-PM-10_00_50
Last ObjectModification:
2019_10_24-AM-11_09_05
Theory : equipollence!!cardinality!
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