Nuprl Lemma : fset-subtype2
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. (s ∈ fset({z:T| z ∈ s} ))
Proof
Definitions occuring in Statement :
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fset: fset(T),
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
fset-member: a ∈ s,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
set-equal: set-equal(T;x;y),
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
fset_wf,
fset-member_wf,
quotient-member-eq,
list_wf,
assert_wf,
deq-member_wf,
set-equal_wf,
set-equal-equiv,
list-subtype,
subtype_rel_list,
l_member_wf,
subtype_rel_sets,
assert-deq-member,
deq_wf,
subtype_rel_list_set,
istype-assert,
sq_stable_from_decidable,
decidable__assert,
l_member-settype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
setEquality,
hypothesisEquality,
sqequalRule,
hypothesis,
pertypeElimination,
promote_hyp,
productElimination,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
Error :universeIsType,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
applyEquality,
setElimination,
rename,
Error :setIsType,
Error :lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
Error :productIsType,
Error :equalityIstype,
sqequalBase,
axiomEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
Error :dependent_set_memberEquality_alt
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. (s \mmember{} fset(\{z:T| z \mmember{} s\} ))
Date html generated:
2019_06_20-PM-01_58_39
Last ObjectModification:
2018_12_19-PM-05_04_02
Theory : finite!sets
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