Nuprl Lemma : setmem-set-part
∀s,x:coSet{i:l}. ((x ∈ set-part(s)) ⇐⇒ (x ∈ s) ∧ isSet(x))
Proof
Definitions occuring in Statement :
set-part: set-part(s),
isSet: isSet(w),
setmem: (x ∈ s),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
set-part: set-part(s),
all: ∀x:A. B[x]
Lemmas referenced :
iff_wf,
sub-set_wf,
isSet-set-predicate,
coSet_wf,
setmem-sub-coset,
isSet_wf,
setmem_wf
Rules used in proof :
independent_functionElimination,
cumulativity,
setEquality,
rename,
setElimination,
lambdaEquality,
sqequalRule,
dependent_functionElimination,
impliesFunctionality,
addLevel,
because_Cache,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
productEquality,
hypothesis,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s,x:coSet\{i:l\}. ((x \mmember{} set-part(s)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} s) \mwedge{} isSet(x))
Date html generated:
2018_07_29-AM-09_52_41
Last ObjectModification:
2018_07_25-PM-03_55_54
Theory : constructive!set!theory
Home
Index