Nuprl Lemma : equipollent-singleton-domain
∀[S:Type]. ∀s:singleton-type(S). ∀[A:S ⟶ Type]. u:S ⟶ (A u) ~ A (fst(s))
Proof
Definitions occuring in Statement :
singleton-type: singleton-type(A),
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
singleton-type: singleton-type(A),
exists: ∃x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
pi1: fst(t),
equipollent: A ~ B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f)
Lemmas referenced :
equipollent_inversion,
singleton-type_wf,
istype-universe,
subtype_rel-equal,
equal_wf,
iff_weakening_equal,
biject_wf,
squash_wf,
true_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
independent_pairEquality,
inhabitedIsType,
hypothesis,
sqequalRule,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
functionIsType,
universeIsType,
instantiate,
universeEquality,
dependent_pairFormation_alt,
lambdaEquality_alt,
independent_isectElimination,
imageElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation,
applyLambdaEquality,
functionExtensionality_alt
Latex:
\mforall{}[S:Type]. \mforall{}s:singleton-type(S). \mforall{}[A:S {}\mrightarrow{} Type]. u:S {}\mrightarrow{} (A u) \msim{} A (fst(s))
Date html generated:
2020_05_19-PM-10_00_28
Last ObjectModification:
2020_01_04-PM-08_00_35
Theory : equipollence!!cardinality!
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