Nuprl Lemma : equipollent-empty-domain
∀[S:Type]. ((¬S) ⇒ (∀[A:S ⟶ Type]. u:S ⟶ (A u) ~ ℕ1))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
false: False,
not: ¬A,
exists: ∃x:A. B[x],
singleton-type: singleton-type(A),
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
equal_wf,
all_wf,
not_wf,
singleton-type-one,
int_seg_wf,
equipollent-singletons
Rules used in proof :
lambdaEquality,
voidEquality,
isect_memberEquality,
voidElimination,
functionExtensionality,
dependent_pairFormation,
sqequalRule,
universeEquality,
cumulativity,
independent_functionElimination,
hypothesis,
natural_numberEquality,
applyEquality,
hypothesisEquality,
functionEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[S:Type]. ((\mneg{}S) {}\mRightarrow{} (\mforall{}[A:S {}\mrightarrow{} Type]. u:S {}\mrightarrow{} (A u) \msim{} \mBbbN{}1))
Date html generated:
2018_07_29-AM-09_23_21
Last ObjectModification:
2018_07_27-PM-05_25_54
Theory : equipollence!!cardinality!
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