Nuprl Lemma : in-open-union
∀[X:Type]. ∀[x:X]. ∀[A:ℕ ⟶ Open(X)]. (x ∈ open-union(n.A[n]) ⇐⇒ ¬¬(∃n:ℕ. ((A[n] x) = ⊤ ∈ Sierpinski)))
Proof
Definitions occuring in Statement :
in-open: x ∈ A,
open-union: open-union(n.A[n]),
Open: Open(X),
Sierpinski: Sierpinski,
Sierpinski-top: ⊤,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
open-union: open-union(n.A[n]),
in-open: x ∈ A,
Open: Open(X),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
not_wf,
exists_wf,
nat_wf,
equal_wf,
Sierpinski_wf,
Open_wf,
Sierpinski-top_wf,
subtype-Sierpinski,
in-open_wf,
open-union_wf,
sp-lub-is-top
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
independent_functionElimination,
voidElimination,
lemma_by_obid,
isectElimination,
lambdaEquality,
applyEquality,
hypothesisEquality,
because_Cache,
productElimination,
independent_pairEquality,
dependent_functionElimination,
axiomEquality,
functionEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[x:X]. \mforall{}[A:\mBbbN{} {}\mrightarrow{} Open(X)]. (x \mmember{} open-union(n.A[n]) \mLeftarrow{}{}\mRightarrow{} \mneg{}\mneg{}(\mexists{}n:\mBbbN{}. ((A[n] x) = \mtop{})))
Date html generated:
2019_10_31-AM-07_19_02
Last ObjectModification:
2015_12_28-AM-11_21_48
Theory : synthetic!topology
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