Nuprl Lemma : setmem-productset
∀a,b,p:coSet{i:l}. ((p ∈ a x b) ⇐⇒ ∃u,v:coSet{i:l}. ((u ∈ a) ∧ (v ∈ b) ∧ seteq(p;(u,v))))
Proof
Definitions occuring in Statement :
productset: a x b,
orderedpairset: (a,b),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
orderedpairset: (a,b),
cand: A c∧ B,
exists: ∃x:A. B[x],
top: Top,
productset: a x b,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
mk-coset: mk-coset(T;f),
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
singleset_functionality,
seteq_weakening,
pairset_functionality,
singleset_wf,
pairset_wf,
seteq_functionality,
setmem-coset,
setmem-mk-coset,
orderedpairset_wf,
seteq_wf,
coSet_wf,
exists_wf,
mk-coset_wf,
productset_wf,
setmem_wf,
coSet_subtype,
subtype_coSet
Rules used in proof :
independent_functionElimination,
spreadEquality,
independent_pairEquality,
dependent_functionElimination,
dependent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
because_Cache,
cumulativity,
productEquality,
lambdaEquality,
instantiate,
isectElimination,
independent_pairFormation,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
hypothesis,
extract_by_obid,
introduction,
cut,
hypothesis_subsumption,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,b,p:coSet\{i:l\}. ((p \mmember{} a x b) \mLeftarrow{}{}\mRightarrow{} \mexists{}u,v:coSet\{i:l\}. ((u \mmember{} a) \mwedge{} (v \mmember{} b) \mwedge{} seteq(p;(u,v))))
Date html generated:
2018_07_29-AM-10_04_03
Last ObjectModification:
2018_07_18-PM-11_45_25
Theory : constructive!set!theory
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