Nuprl Lemma : assert-fset-disjoint
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:fset(T)]. uiff(↑fset-disjoint(eq;as;bs);∀x:T. (¬(x ∈ as ∧ x ∈ bs)))
Proof
Definitions occuring in Statement :
fset-disjoint: fset-disjoint(eq;as;bs),
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,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
fset-disjoint: fset-disjoint(eq;as;bs),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
fset-member: a ∈ s,
assert: ↑b,
ifthenelse: if b then t else f fi ,
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
it: ⋅,
bfalse: ff,
top: Top
Lemmas referenced :
fset-member_wf,
equal-wf-T-base,
fset_wf,
fset-intersection_wf,
all_wf,
not_wf,
iff_weakening_uiff,
assert_wf,
fset-null_wf,
assert-fset-null,
assert_witness,
uiff_wf,
deq_wf,
member-fset-intersection,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
fset-member_witness,
empty-fset_wf,
fset-extensionality,
member-empty-fset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
independent_pairFormation,
introduction,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
productEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
because_Cache,
baseClosed,
addLevel,
productElimination,
independent_isectElimination,
cumulativity,
universeEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
instantiate,
isect_memberEquality,
independent_pairEquality,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:fset(T)].
uiff(\muparrow{}fset-disjoint(eq;as;bs);\mforall{}x:T. (\mneg{}(x \mmember{} as \mwedge{} x \mmember{} bs)))
Date html generated:
2019_06_20-PM-01_59_06
Last ObjectModification:
2018_08_24-PM-11_34_40
Theory : finite!sets
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