Nuprl Lemma : equipollent-split
∀[T:Type]. ∀[P:T ⟶ ℙ]. ((∀x:T. Dec(↓P[x])) ⇒ T ~ {x:T| P[x]} + {x:T| ¬P[x]} )
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
false: False,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
uimplies: b supposing a,
guard: {T},
isl: isl(x),
uiff: uiff(P;Q)
Lemmas referenced :
decidable_wf,
squash_wf,
istype-universe,
subtype_rel_self,
not_wf,
equipollent_functionality_wrt_equipollent2,
union_functionality_wrt_equipollent,
equipollent-set,
biject_wf,
equal_functionality_wrt_subtype_rel2,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
not_squash
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalRule,
functionIsType,
universeIsType,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesis,
universeEquality,
instantiate,
unionEquality,
setEquality,
because_Cache,
lambdaEquality_alt,
independent_functionElimination,
productElimination,
rename,
dependent_pairFormation_alt,
inhabitedIsType,
unionElimination,
inlEquality_alt,
dependent_set_memberEquality_alt,
setIsType,
inrEquality_alt,
imageMemberEquality,
baseClosed,
voidElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_pairFormation,
unionIsType,
imageElimination,
applyLambdaEquality,
setElimination,
independent_isectElimination,
productIsType
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:T. Dec(\mdownarrow{}P[x])) {}\mRightarrow{} T \msim{} \{x:T| P[x]\} + \{x:T| \mneg{}P[x]\} )
Date html generated:
2020_05_19-PM-10_00_26
Last ObjectModification:
2020_01_04-PM-08_00_41
Theory : equipollence!!cardinality!
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