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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