Nuprl Lemma : ss-mem-open-and
∀[X:SeparationSpace]. ∀A,B:Open(X). ∀x:Point(X). (x ∈ A ⋂ B ⇐⇒ x ∈ A ∧ x ∈ B)
Proof
Definitions occuring in Statement :
ss-open-and: A ⋂ B,
ss-mem-open: x ∈ O,
ss-open: Open(X),
ss-point: Point(ss),
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
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-and: A ⋂ B,
ss-mem-open: x ∈ O,
exists: ∃x:A. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
uiff: uiff(P;Q),
ss-open: Open(X),
rev_implies: P ⇐ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ss-mem-basic_wf,
squash_wf,
true_wf,
ss-point_wf,
ss-basic_wf,
subtype_rel_self,
iff_weakening_equal,
ss-mem-basic-and,
ss-mem-open_wf,
ss-open-and_wf,
ss-basic-and_wf,
equal_wf,
exists_wf,
ss-open_wf,
separation-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
cut,
applyEquality,
lambdaEquality,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
dependent_pairFormation,
productEquality,
rename
Latex:
\mforall{}[X:SeparationSpace]. \mforall{}A,B:Open(X). \mforall{}x:Point(X). (x \mmember{} A \mcap{} B \mLeftarrow{}{}\mRightarrow{} x \mmember{} A \mwedge{} x \mmember{} B)
Date html generated:
2020_05_20-PM-01_22_37
Last ObjectModification:
2018_07_06-PM-04_56_17
Theory : intuitionistic!topology
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