Nuprl Lemma : ss-mem-open-or
∀[X:SeparationSpace]. ∀T:Type. ∀F:T ⟶ Open(X). ∀x:Point(X). (x ∈ ⋃t:T.F[t] ⇐⇒ ∃t:T. x ∈ F[t])
Proof
Definitions occuring in Statement :
ss-open-or: ⋃x:T.F[x],
ss-mem-open: x ∈ O,
ss-open: Open(X),
ss-point: Point(ss),
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
ss-open-or: ⋃x:T.F[x],
ss-mem-open: x ∈ O,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
ss-open: Open(X),
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
subtype_rel_self,
ss-basic_wf,
ss-mem-basic_wf,
exists_wf,
ss-mem-open_wf,
ss-open-or_wf,
ss-point_wf,
ss-open_wf,
separation-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
hypothesis,
productEquality,
cut,
applyEquality,
instantiate,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
cumulativity,
universeEquality,
lambdaEquality,
because_Cache
Latex:
\mforall{}[X:SeparationSpace]. \mforall{}T:Type. \mforall{}F:T {}\mrightarrow{} Open(X). \mforall{}x:Point(X). (x \mmember{} \mcup{}t:T.F[t] \mLeftarrow{}{}\mRightarrow{} \mexists{}t:T. x \mmember{} F[t])
Date html generated:
2020_05_20-PM-01_22_44
Last ObjectModification:
2018_07_06-PM-05_20_55
Theory : intuitionistic!topology
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