Nuprl Lemma : axiom-choice-C0
∀P:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ ℙ. ((∀f:ℕ ⟶ 𝔹. ⇃(∃m:ℕ. (P m f))) ⇒ ⇃(∃F:(ℕ ⟶ 𝔹) ⟶ ℕ. ∀f:ℕ ⟶ 𝔹. (P (F f) f)))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
cand: A c∧ B,
guard: {T},
pi1: fst(t)
Lemmas referenced :
equal_wf,
implies-quotient-true,
nat-retractible,
bool_subtype_base,
int-value-type,
set-value-type,
int_subtype_base,
le_wf,
set_subtype_base,
canonicalizable-function,
canonicalizable_wf,
implies-prop-truncation,
all-quotient-true,
equiv_rel_true,
true_wf,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
subtype_rel_dep_function,
exists_wf,
quotient_wf,
bool_wf,
nat_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
applyEquality,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
intEquality,
promote_hyp,
dependent_pairFormation,
introduction,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}P:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbP{}
((\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \00D9(\mexists{}m:\mBbbN{}. (P m f))) {}\mRightarrow{} \00D9(\mexists{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. (P (F f) f)))
Date html generated:
2016_05_14-PM-09_42_27
Last ObjectModification:
2016_02_04-PM-03_51_42
Theory : continuity
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