Nuprl Lemma : fset-contains-none-closed-downward
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))].
∀x,y:fset(T). (y ⊆ x ⇒ (↑fset-contains-none(eq;x;a.Cs[a])) ⇒ (↑fset-contains-none(eq;y;a.Cs[a])))
Proof
Definitions occuring in Statement :
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
f-subset: xs ⊆ ys
Lemmas referenced :
f-subset_wf,
fset-member_wf,
fset_wf,
deq-fset_wf,
all_wf,
not_wf,
assert-fset-contains-none,
assert_wf,
fset-contains-none_wf,
deq_wf,
assert_witness,
f-subset_transitivity
Rules used in proof :
cut,
thin,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
addLevel,
impliesFunctionality,
productElimination,
independent_isectElimination,
cumulativity,
universeEquality,
isect_memberFormation,
introduction,
dependent_functionElimination,
isect_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))].
\mforall{}x,y:fset(T).
(y \msubseteq{} x {}\mRightarrow{} (\muparrow{}fset-contains-none(eq;x;a.Cs[a])) {}\mRightarrow{} (\muparrow{}fset-contains-none(eq;y;a.Cs[a])))
Date html generated:
2016_05_14-PM-03_42_27
Last ObjectModification:
2015_12_26-PM-06_39_43
Theory : finite!sets
Home
Index