Nuprl Lemma : sp-lub-is-top1
∀[A:ℕ ⟶ Sierpinski]. (lub(n.A[n]) = ⊤ ∈ Sierpinski ⇐⇒ ¬(∀n:ℕ. (A[n] = ⊥ ∈ Sierpinski)))
Proof
Definitions occuring in Statement :
sp-lub: lub(n.A[n]),
Sierpinski: Sierpinski,
Sierpinski-top: ⊤,
Sierpinski-bottom: ⊥,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
function: x:A ⟶ B[x],
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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x]
Lemmas referenced :
all_wf,
nat_wf,
equal-wf-T-base,
Sierpinski_wf,
sp-lub_wf,
not_wf,
Sierpinski-unequal,
sp-lub-is-bottom,
not-Sierpinski-bottom
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
baseClosed,
because_Cache,
productElimination,
independent_pairEquality,
dependent_functionElimination,
axiomEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
promote_hyp,
addLevel,
impliesFunctionality,
levelHypothesis
Latex:
\mforall{}[A:\mBbbN{} {}\mrightarrow{} Sierpinski]. (lub(n.A[n]) = \mtop{} \mLeftarrow{}{}\mRightarrow{} \mneg{}(\mforall{}n:\mBbbN{}. (A[n] = \mbot{})))
Date html generated:
2019_10_31-AM-06_36_18
Last ObjectModification:
2017_07_28-AM-09_12_09
Theory : synthetic!topology
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