Nuprl Lemma : strong-subtype-fset-member-type
∀[A,B:Type]. ∀[eq:EqDecider(B)]. ∀[s:fset(A)]. ∀[x:B]. x ∈ A supposing x ∈ s supposing strong-subtype(A;B)
Proof
Definitions occuring in Statement :
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
prop: ℙ,
fset-member: a ∈ s,
all: ∀x:A. B[x],
guard: {T},
istype: istype(T),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
fset: fset(T),
quotient: x,y:A//B[x; y]
Lemmas referenced :
fset-subtype,
fset-member_wf,
fset_wf,
strong-subtype_wf,
deq_wf,
istype-universe,
assert-deq-member,
subtype_rel_list,
strong-subtype-l_member-type,
list_wf,
set-equal_wf,
set-equal-reflex,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
hypothesisEquality,
applyEquality,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
independent_isectElimination,
hypothesis,
productElimination,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
Error :inhabitedIsType,
instantiate,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
promote_hyp,
Error :lambdaFormation_alt,
pointwiseFunctionality,
pertypeElimination,
Error :productIsType,
Error :equalityIstype,
sqequalBase
Latex:
\mforall{}[A,B:Type]. \mforall{}[eq:EqDecider(B)].
\mforall{}[s:fset(A)]. \mforall{}[x:B]. x \mmember{} A supposing x \mmember{} s supposing strong-subtype(A;B)
Date html generated:
2019_06_20-PM-01_58_36
Last ObjectModification:
2018_11_24-PM-06_40_37
Theory : finite!sets
Home
Index