Nuprl Lemma : assert-fset-contains-none
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. ∀[Cs:T ⟶ fset(fset(T))].
uiff(↑fset-contains-none(eq;s;x.Cs[x]);∀x:T. (x ∈ s ⇒ (∀c:fset(T). (c ∈ Cs[x] ⇒ (¬c ⊆ s)))))
Proof
Definitions occuring in Statement :
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
deq-fset: deq-fset(eq),
f-subset: xs ⊆ ys,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
squash: ↓T
Lemmas referenced :
member-f-union,
exists_wf,
squash_wf,
and_wf,
deq_wf,
fset-contains-none_wf,
uiff_wf,
assert_witness,
rev_implies_wf,
assert-fset-contains-none-of,
f-subset_wf,
not_wf,
fset-member_wf,
all_wf,
deq-fset_wf,
fset_wf,
f-union_wf,
fset-contains-none-of_wf,
assert_wf,
iff_weakening_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
addLevel,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
isect_memberFormation,
introduction,
independent_isectElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
independent_functionElimination,
dependent_functionElimination,
voidElimination,
cumulativity,
because_Cache,
universeEquality,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
allFunctionality,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))].
uiff(\muparrow{}fset-contains-none(eq;s;x.Cs[x]);\mforall{}x:T. (x \mmember{} s {}\mRightarrow{} (\mforall{}c:fset(T). (c \mmember{} Cs[x] {}\mRightarrow{} (\mneg{}c \msubseteq{} s)))))
Date html generated:
2016_05_14-PM-03_42_21
Last ObjectModification:
2016_01_14-PM-10_41_03
Theory : finite!sets
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