Nuprl Lemma : equipollent-set
∀[T:Type]. ∀[P:T ⟶ ℙ]. {x:T| P[x]} ~ {x:T| ↓P[x]}
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
squash: ↓T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
surject: Surj(A;B;f),
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
inject: Inj(A;B;f),
and: P ∧ Q,
biject: Bij(A;B;f),
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
squash: ↓T,
member: t ∈ T,
exists: ∃x:A. B[x],
equipollent: A ~ B,
uall: ∀[x:A]. B[x]
Lemmas referenced :
biject_wf,
set_wf,
equal_wf,
member_wf,
squash_wf
Rules used in proof :
functionEquality,
imageElimination,
because_Cache,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
lambdaFormation,
independent_pairFormation,
universeEquality,
setEquality,
cumulativity,
functionExtensionality,
applyEquality,
isectElimination,
extract_by_obid,
sqequalHypSubstitution,
baseClosed,
imageMemberEquality,
sqequalRule,
introduction,
hypothesis,
hypothesisEquality,
dependent_set_memberEquality,
cut,
rename,
thin,
setElimination,
lambdaEquality,
dependent_pairFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \{x:T| P[x]\} \msim{} \{x:T| \mdownarrow{}P[x]\}
Date html generated:
2018_05_21-PM-00_52_43
Last ObjectModification:
2017_12_07-PM-06_30_56
Theory : equipollence!!cardinality!
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