Nuprl Lemma : fset-ac-le-implies
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[ac1,ac2:fset(fset(T))].
(fset-ac-le(eq;ac1;ac2) ⇒ {∀a:fset(T). (a ∈ ac1 ⇒ (¬({y ∈ ac2 | deq-f-subset(eq) y a} = {} ∈ fset(fset(T)))))})
Proof
Definitions occuring in Statement :
fset-ac-le: fset-ac-le(eq;ac1;ac2),
deq-fset: deq-fset(eq),
deq-f-subset: deq-f-subset(eq),
empty-fset: {},
fset-filter: {x ∈ s | P[x]},
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
fset-ac-le: fset-ac-le(eq;ac1;ac2),
uiff: uiff(P;Q),
uimplies: b supposing a
Lemmas referenced :
equal-wf-T-base,
fset_wf,
fset-filter_wf,
deq-f-subset_wf,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
assert_wf,
fset-member_wf,
deq-fset_wf,
fset-ac-le_wf,
deq_wf,
fset-all-iff,
bnot_wf,
fset-null_wf,
assert_of_bnot,
assert-fset-null
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
functionEquality,
functionExtensionality,
baseClosed,
dependent_functionElimination,
because_Cache,
isect_memberEquality,
universeEquality,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp
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{} (\mneg{}(\{y \mmember{} ac2 | deq-f-subset(eq) y a\} = \{\})))\})
Date html generated:
2017_04_17-AM-09_20_36
Last ObjectModification:
2017_02_27-PM-05_23_24
Theory : finite!sets
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