Nuprl Lemma : fset-ac-le-iff
∀[T:Type]
∀eq:EqDecider(T). ∀ac1,ac2:fset(fset(T)).
(fset-ac-le(eq;ac1;ac2) ⇐⇒ ∀[a:fset(T)]. ↓∃b:fset(T). (b ∈ ac2 ∧ b ⊆ a) supposing a ∈ ac1)
Proof
Definitions occuring in Statement :
fset-ac-le: fset-ac-le(eq;ac1;ac2),
deq-fset: deq-fset(eq),
f-subset: xs ⊆ ys,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
fset-ac-le: fset-ac-le(eq;ac1;ac2),
ac-covers: ac-covers(eq;ac;x),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
squash: ↓T,
fset-all: fset-all(s;x.P[x]),
subtype_rel: A ⊆r B
Lemmas referenced :
fset-all-iff,
fset_wf,
deq-fset_wf,
ac-covers_wf,
fset-all_wf,
uall_wf,
isect_wf,
fset-member_wf,
squash_wf,
exists_wf,
f-subset_wf,
iff_wf,
assert_wf,
deq_wf,
fset-ac-le_wf,
assert_witness,
fset-null_wf,
fset-filter_wf,
bnot_wf,
deq-f-subset_wf,
bool_wf,
all_wf,
iff_weakening_uiff,
assert-ac-covers
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
addLevel,
productElimination,
independent_pairFormation,
independent_functionElimination,
lambdaEquality,
independent_isectElimination,
productEquality,
dependent_functionElimination,
independent_pairEquality,
isect_memberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setElimination,
rename,
setEquality,
functionEquality,
because_Cache,
universeEquality,
cumulativity,
promote_hyp,
isectEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}ac1,ac2:fset(fset(T)).
(fset-ac-le(eq;ac1;ac2) \mLeftarrow{}{}\mRightarrow{} \mforall{}[a:fset(T)]. \mdownarrow{}\mexists{}b:fset(T). (b \mmember{} ac2 \mwedge{} b \msubseteq{} a) supposing a \mmember{} ac1)
Date html generated:
2019_06_20-PM-01_59_33
Last ObjectModification:
2018_08_24-PM-11_40_43
Theory : finite!sets
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