Nuprl Lemma : assert-fset-contains-none-of
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. ∀[cs:fset(fset(T))].
uiff(↑fset-contains-none-of(eq;s;cs);∀c:fset(T). (c ∈ cs ⇒ (¬c ⊆ s)))
Proof
Definitions occuring in Statement :
fset-contains-none-of: fset-contains-none-of(eq;s;cs),
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],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
fset-contains-none-of: fset-contains-none-of(eq;s;cs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
fset-filter-is-empty,
deq-fset_wf,
assert-deq-f-subset,
fset-member_wf,
fset_wf,
assert_wf,
deq-f-subset_wf,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
not_wf,
exists_wf,
iff_weakening_uiff,
equal-wf-T-base,
uiff_wf,
fset-null_wf,
fset-filter_wf,
assert-fset-null,
assert_witness,
fset-contains-none-of_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
cumulativity,
hypothesisEquality,
hypothesis,
independent_pairFormation,
isect_memberFormation,
lambdaFormation,
independent_functionElimination,
dependent_pairFormation,
productElimination,
independent_isectElimination,
productEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
functionEquality,
sqequalRule,
functionExtensionality,
voidElimination,
dependent_functionElimination,
addLevel,
instantiate,
baseClosed,
universeEquality,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. \mforall{}[cs:fset(fset(T))].
uiff(\muparrow{}fset-contains-none-of(eq;s;cs);\mforall{}c:fset(T). (c \mmember{} cs {}\mRightarrow{} (\mneg{}c \msubseteq{} s)))
Date html generated:
2017_04_17-AM-09_20_26
Last ObjectModification:
2017_02_27-PM-05_23_34
Theory : finite!sets
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