Nuprl Lemma : fset-some-iff2
∀[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],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
fset-some: fset-some(s;x.P[x]),
exists: ∃x:A. B[x]
Lemmas referenced :
fset-some-iff,
all_wf,
fset-member_wf,
not_wf,
assert_wf,
fset-some_wf,
exists_wf,
and_wf,
fset-null_wf,
fset-filter_wf,
fset_wf,
bool_wf,
deq_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
productElimination,
introduction,
independent_isectElimination,
lambdaFormation,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
dependent_functionElimination,
universeEquality,
dependent_pairFormation
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{}(\mforall{}x:T. (x \mmember{} s {}\mRightarrow{} (\mneg{}\muparrow{}P[x]))))
Date html generated:
2016_05_14-PM-03_41_07
Last ObjectModification:
2015_12_26-PM-06_40_34
Theory : finite!sets
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