Nuprl Lemma : equipollent-length
∀[T:Type]. ∀L:T List. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ {x:T| (x ∈ L)} ~ ℕ||L|| supposing no_repeats(T;L))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
length: ||as||,
list: T List,
int_seg: {i..j-},
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
list_wf,
decidable_wf,
all_wf,
equal_wf,
no_repeats_wf,
and_wf,
set_wf,
sq_stable__l_member,
l_member-set,
length_wf,
no_repeats-subtype,
list-subtype,
length_wf_nat,
l_member_wf,
equipollent-iff-list,
no_repeats_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
setEquality,
dependent_functionElimination,
productElimination,
dependent_pairFormation,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality,
setElimination,
because_Cache,
independent_pairFormation,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
intEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} \{x:T| (x \mmember{} L)\} \msim{} \mBbbN{}||L|| supposing no\_repeats(T;L))
Date html generated:
2016_05_14-PM-04_03_43
Last ObjectModification:
2016_01_14-PM-11_05_29
Theory : equipollence!!cardinality!
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