Nuprl Lemma : finite-type-implies-finite
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ finite-type(T) ⇒ finite(T))
Proof
Definitions occuring in Statement :
finite: finite(T),
finite-type: finite-type(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
prop: ℙ,
nat: ℕ
Lemmas referenced :
finite-type-equipollent,
finite-type_wf,
all_wf,
decidable_wf,
equal_wf,
finite_functionality_wrt_equipollent,
int_seg_wf,
nsub_finite
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
productElimination,
sqequalRule,
lambdaEquality,
universeEquality,
natural_numberEquality,
setElimination,
rename,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} finite-type(T) {}\mRightarrow{} finite(T))
Date html generated:
2016_05_14-PM-04_05_29
Last ObjectModification:
2015_12_26-PM-07_41_38
Theory : equipollence!!cardinality!
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