Nuprl Lemma : finite-decidable-subset
∀T:Type. ∀B:T ⟶ ℙ. (finite(T) ⇒ (∀x:T. Dec(↓B[x])) ⇒ finite({t:T| B[t]} ))
Proof
Definitions occuring in Statement :
finite: finite(T),
decidable: Dec(P),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equipollent-split,
all_wf,
decidable_wf,
squash_wf,
finite_wf,
finite_functionality_wrt_equipollent,
not_wf,
finite-union
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
unionEquality,
setEquality,
productElimination,
dependent_functionElimination,
independent_pairFormation
Latex:
\mforall{}T:Type. \mforall{}B:T {}\mrightarrow{} \mBbbP{}. (finite(T) {}\mRightarrow{} (\mforall{}x:T. Dec(\mdownarrow{}B[x])) {}\mRightarrow{} finite(\{t:T| B[t]\} ))
Date html generated:
2016_10_21-AM-11_00_59
Last ObjectModification:
2016_08_06-PM-04_52_14
Theory : equipollence!!cardinality!
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