Nuprl Lemma : ss-mem-whole
∀X:SeparationSpace. ∀x:Point(X). uiff(x ∈ ss-whole(X);True)
Proof
Definitions occuring in Statement :
ss-whole: ss-whole(X),
ss-mem-open: x ∈ O,
ss-point: Point(ss),
separation-space: SeparationSpace,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
true: True
Definitions unfolded in proof :
all: ∀x:A. B[x],
ss-whole: ss-whole(X),
ss-mem-open: x ∈ O,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
true: True,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
cand: A c∧ B
Lemmas referenced :
exists_wf,
ss-basic_wf,
equal-wf-T-base,
ss-mem-basic_wf,
ss-basic-whole_wf,
ss-mem-basic-whole,
true_wf,
ss-point_wf,
separation-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
natural_numberEquality,
sqequalHypSubstitution,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
productEquality,
baseClosed,
rename,
dependent_pairFormation,
because_Cache,
dependent_functionElimination,
productElimination,
independent_isectElimination
Latex:
\mforall{}X:SeparationSpace. \mforall{}x:Point(X). uiff(x \mmember{} ss-whole(X);True)
Date html generated:
2020_05_20-PM-01_22_30
Last ObjectModification:
2018_07_06-PM-02_24_08
Theory : intuitionistic!topology
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