Nuprl Lemma : finite-iff-listable
∀[T:Type]. (finite(T) ⇐⇒ ∃L:T List. (no_repeats(T;L) ∧ (∀x:T. (x ∈ L))))
Proof
Definitions occuring in Statement :
finite: finite(T),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
finite: finite(T),
exists: ∃x:A. B[x],
cand: A c∧ B,
nat: ℕ
Lemmas referenced :
finite_wf,
exists_wf,
list_wf,
no_repeats_wf,
all_wf,
l_member_wf,
equipollent-iff-list,
length_wf_nat,
equipollent_wf,
int_seg_wf,
length_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
universeEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
natural_numberEquality,
setElimination,
rename,
intEquality
Latex:
\mforall{}[T:Type]. (finite(T) \mLeftarrow{}{}\mRightarrow{} \mexists{}L:T List. (no\_repeats(T;L) \mwedge{} (\mforall{}x:T. (x \mmember{} L))))
Date html generated:
2017_04_17-AM-09_33_46
Last ObjectModification:
2017_02_27-PM-05_32_47
Theory : equipollence!!cardinality!
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