Nuprl Lemma : seteq-orderedpairs-iff
∀a,b,a',b':coSet{i:l}. (seteq((a,b);(a',b')) ⇐⇒ seteq(a;a') ∧ seteq(b;b'))
Proof
Definitions occuring in Statement :
orderedpairset: (a,b),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
cand: A c∧ B,
guard: {T},
or: P ∨ Q,
orderedpairset: (a,b),
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
singleset_functionality,
pairset_functionality,
seteq_transitivity,
seteq_functionality,
seteq_inversion,
seteq_weakening,
iff_wf,
setmem_wf,
setmem-singleset,
or_wf,
pairset_wf,
singleset_wf,
setmem-pairset,
co-seteq-iff,
coSet_wf,
orderedpairset_wf,
seteq_wf
Rules used in proof :
orFunctionality,
inrFormation,
sqequalRule,
unionElimination,
impliesFunctionality,
addLevel,
inlFormation,
promote_hyp,
allFunctionality,
independent_functionElimination,
productElimination,
dependent_functionElimination,
because_Cache,
productEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,b,a',b':coSet\{i:l\}. (seteq((a,b);(a',b')) \mLeftarrow{}{}\mRightarrow{} seteq(a;a') \mwedge{} seteq(b;b'))
Date html generated:
2018_07_29-AM-09_59_46
Last ObjectModification:
2018_07_18-AM-11_14_34
Theory : constructive!set!theory
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