Nuprl Lemma : strong-subtype-iff-preserves-singleton
∀[A,B:Type]. uiff(strong-subtype(A;B);(A ⊆r B) ∧ (∀a:A. ({a:B} ⊆r {a:A})))
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
singleton: {a:T},
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
strong-subtype: strong-subtype(A;B),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
cand: A c∧ B,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
singleton: {a:T},
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x]
Lemmas referenced :
singleton_properties,
singleton_wf,
subtype_rel_wf,
exists_wf,
equal_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
lambdaFormation,
lambdaEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
setElimination,
rename,
cumulativity,
applyEquality,
independent_pairEquality,
axiomEquality,
dependent_functionElimination,
productEquality,
setEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_set_memberEquality,
dependent_pairFormation
Latex:
\mforall{}[A,B:Type]. uiff(strong-subtype(A;B);(A \msubseteq{}r B) \mwedge{} (\mforall{}a:A. (\{a:B\} \msubseteq{}r \{a:A\})))
Date html generated:
2017_04_14-AM-07_36_44
Last ObjectModification:
2017_02_27-PM-03_09_29
Theory : subtype_1
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