Nuprl Lemma : fset-filter-is-empty
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:T ⟶ 𝔹]. ∀[s:fset(T)].
uiff({x ∈ s | P[x]} = {} ∈ fset(T);¬(∃x:T. (x ∈ s ∧ (↑P[x]))))
Proof
Definitions occuring in Statement :
empty-fset: {},
fset-filter: {x ∈ s | P[x]},
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
all: ∀x:A. B[x],
cand: A c∧ B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-universe,
fset-member_wf,
assert_wf,
fset-filter_wf,
not_wf,
exists_wf,
fset_wf,
bool_wf,
deq_wf,
mem_empty_lemma,
member-fset-filter,
fset-extensionality,
empty-fset_wf,
istype-void,
fset-member_witness,
assert_witness,
iff_weakening_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
Error :lambdaFormation_alt,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
sqequalRule,
Error :productIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
Error :universeIsType,
applyEquality,
Error :lambdaEquality_alt,
dependent_functionElimination,
because_Cache,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :equalityIsType3,
baseClosed,
productEquality,
productElimination,
independent_pairEquality,
Error :isect_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
Error :functionIsType,
universeEquality,
functionExtensionality,
lambdaEquality,
cumulativity,
lambdaFormation,
voidEquality,
isect_memberEquality,
applyLambdaEquality,
hyp_replacement,
independent_isectElimination,
promote_hyp,
Error :dependent_pairFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)].
uiff(\{x \mmember{} s | P[x]\} = \{\};\mneg{}(\mexists{}x:T. (x \mmember{} s \mwedge{} (\muparrow{}P[x]))))
Date html generated:
2019_06_20-PM-01_59_17
Last ObjectModification:
2018_10_06-PM-11_55_35
Theory : finite!sets
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