Nuprl Lemma : setmem-iff
∀x,s:coSet{i:l}. ((x ∈ s) ⇐⇒ ∃t:set-dom(s). seteq(x;set-item(s;t)))
Proof
Definitions occuring in Statement :
setmem: (x ∈ s),
seteq: seteq(s1;s2),
set-item: set-item(s;x),
set-dom: set-dom(s),
coSet: coSet{i:l},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
seteq: seteq(s1;s2),
set-dom: set-dom(s),
set-item: set-item(s;x),
pi2: snd(t),
pi1: fst(t),
coW-dom: coW-dom(a.B[a];w),
coW-item: coW-item(w;b),
coWmem: coWmem(a.B[a];z;w),
setmem: (x ∈ s),
subtype_rel: A ⊆r B,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
coSet_wf,
seteq_wf,
exists_wf,
coSet_subtype,
subtype_coSet
Rules used in proof :
because_Cache,
lambdaEquality,
isectElimination,
independent_pairFormation,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
hypothesis,
extract_by_obid,
introduction,
cut,
hypothesis_subsumption,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x,s:coSet\{i:l\}. ((x \mmember{} s) \mLeftarrow{}{}\mRightarrow{} \mexists{}t:set-dom(s). seteq(x;set-item(s;t)))
Date html generated:
2018_07_29-AM-09_50_01
Last ObjectModification:
2018_07_11-PM-00_20_54
Theory : constructive!set!theory
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