Nuprl Lemma : fset-some-iff
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:T ⟶ 𝔹]. ∀[s:fset(T)]. uiff(fset-some(s;x.P[x]);¬¬(∃x:T. (x ∈ s ∧ (↑P[x]))))
Proof
Definitions occuring in Statement :
fset-some: fset-some(s;x.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
Definitions unfolded in proof :
fset-some: fset-some(s;x.P[x]),
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
exists: ∃x:A. B[x],
guard: {T},
all: ∀x:A. B[x],
top: Top,
cand: A c∧ B
Lemmas referenced :
assert-fset-null,
fset-filter_wf,
not_wf,
exists_wf,
fset-member_wf,
assert_wf,
fset-null_wf,
fset_wf,
bool_wf,
deq_wf,
fset-extensionality,
empty-fset_wf,
fset-member_witness,
member-fset-filter,
mem_empty_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
sqequalHypSubstitution,
independent_functionElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
productElimination,
independent_isectElimination,
voidElimination,
because_Cache,
productEquality,
dependent_functionElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
dependent_pairFormation,
voidEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)].
uiff(fset-some(s;x.P[x]);\mneg{}\mneg{}(\mexists{}x:T. (x \mmember{} s \mwedge{} (\muparrow{}P[x]))))
Date html generated:
2016_10_21-AM-10_44_40
Last ObjectModification:
2016_07_12-AM-05_51_35
Theory : finite!sets
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