Nuprl Lemma : equipollent-singleton-domain2
∀[S:Type]. (singleton-type(S) ⇒ (∀[A:Type]. S ⟶ A ~ A))
Proof
Definitions occuring in Statement :
singleton-type: singleton-type(A),
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
singleton-type: singleton-type(A),
exists: ∃x:A. B[x],
member: t ∈ T,
equipollent: A ~ B,
prop: ℙ,
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
surject: Surj(A;B;f),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equipollent_inversion,
singleton-type_wf,
istype-universe,
biject_wf,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
functionEquality,
independent_functionElimination,
hypothesis,
inhabitedIsType,
universeIsType,
instantiate,
universeEquality,
dependent_pairFormation_alt,
lambdaEquality_alt,
independent_pairFormation,
sqequalRule,
equalityIstype,
functionIsType,
because_Cache,
applyLambdaEquality,
applyEquality,
functionExtensionality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[S:Type]. (singleton-type(S) {}\mRightarrow{} (\mforall{}[A:Type]. S {}\mrightarrow{} A \msim{} A))
Date html generated:
2020_05_19-PM-10_00_30
Last ObjectModification:
2020_01_04-PM-08_00_38
Theory : equipollence!!cardinality!
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