Nuprl Lemma : strong-continuity2-implies-weak-skolem-cantor
∀F:(ℕ ⟶ 𝔹) ⟶ 𝔹. ⇃(∃M:(ℕ ⟶ 𝔹) ⟶ ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕM f ⟶ 𝔹)) ⇒ F f = F g))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
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,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
pi1: fst(t),
isl: isl(x),
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
iff: P ⇐⇒ Q,
outl: outl(x),
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
quotient: x,y:A//B[x; y],
squash: ↓T
Lemmas referenced :
nat_wf,
bool_wf,
strong-continuity2-no-inner-squash-cantor4,
exists_wf,
int_seg_wf,
unit_wf2,
all_wf,
equal_wf,
subtype_rel_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
isect_wf,
assert_wf,
isl_wf,
pi1_wf,
and_wf,
btrue_wf,
subtype_base_sq,
bool_subtype_base,
iff_imp_equal_bool,
outl_wf,
assert_elim,
true_wf,
quotient_wf,
equiv_rel_true,
quotient-member-eq,
equal-wf-base,
member_wf,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
natural_numberEquality,
setElimination,
rename,
unionEquality,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
because_Cache,
independent_isectElimination,
independent_pairFormation,
inlEquality,
dependent_pairFormation,
functionExtensionality,
independent_pairEquality,
dependent_pairEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberEquality,
applyLambdaEquality,
instantiate,
cumulativity,
promote_hyp,
hyp_replacement,
addLevel,
levelHypothesis,
pointwiseFunctionality,
pertypeElimination,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbB{}. \00D9(\mexists{}M:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((f = g) {}\mRightarrow{} F f = F g))
Date html generated:
2018_05_21-PM-01_19_18
Last ObjectModification:
2018_05_19-AM-06_33_00
Theory : continuity
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