Nuprl Lemma : setmem-plus-set
∀a,x:coSet{i:l}. ((x ∈ (a)+) ⇐⇒ (x ∈ a) ∨ seteq(x;a))
Proof
Definitions occuring in Statement :
plus-set: (a)+,
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
guard: {T},
or: P ∨ Q,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
plus-set: (a)+,
all: ∀x:A. B[x]
Lemmas referenced :
coSet_wf,
iff_wf,
set-add_wf,
setmem-singleset,
singleset_wf,
setmem-set-add,
or_wf,
seteq_wf,
setmem_wf
Rules used in proof :
impliesFunctionality,
orFunctionality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
addLevel,
because_Cache,
inlFormation,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
inrFormation,
hypothesis,
sqequalRule,
thin,
unionElimination,
sqequalHypSubstitution,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,x:coSet\{i:l\}. ((x \mmember{} (a)+) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} a) \mvee{} seteq(x;a))
Date html generated:
2018_07_29-AM-10_00_12
Last ObjectModification:
2018_07_18-PM-01_33_43
Theory : constructive!set!theory
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