Nuprl Lemma : fset-ac-le-implies2
∀[T:Type]
∀eq:EqDecider(T). ∀ac1,ac2:fset(fset(T)).
(fset-ac-le(eq;ac1;ac2) ⇒ {∀a:fset(T). (a ∈ ac1 ⇒ (↓∃b:fset(T). (b ∈ ac2 ∧ b ⊆ a)))})
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),
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
squash: ↓T,
prop: ℙ
Lemmas referenced :
deq_wf,
fset-ac-le_wf,
deq-fset_wf,
fset_wf,
fset-member_wf,
fset-ac-le-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
productElimination,
independent_functionElimination,
independent_isectElimination,
imageElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
lambdaEquality,
because_Cache,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}ac1,ac2:fset(fset(T)).
(fset-ac-le(eq;ac1;ac2) {}\mRightarrow{} \{\mforall{}a:fset(T). (a \mmember{} ac1 {}\mRightarrow{} (\mdownarrow{}\mexists{}b:fset(T). (b \mmember{} ac2 \mwedge{} b \msubseteq{} a)))\})
Date html generated:
2016_05_14-PM-03_43_15
Last ObjectModification:
2016_01_20-PM-02_32_52
Theory : finite!sets
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