Nuprl Lemma : cosetTC-unique
∀a,s:coSet{i:l}.
((a ⊆ s)
⇒ transitive-set(s)
⇒ (∀s':coSet{i:l}. ((a ⊆ s') ⇒ transitive-set(s') ⇒ (s ⊆ s')))
⇒ seteq(s;cosetTC(a)))
Proof
Definitions occuring in Statement :
transitive-set: transitive-set(s),
setsubset: (a ⊆ b),
cosetTC: cosetTC(a),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
cand: A c∧ B,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
transitive-set_wf,
setsubset_wf,
coSet_wf,
all_wf,
cosetTC-least,
cosetTC-transitive,
cosetTC-contains,
cosetTC_wf,
seteq-iff-setsubset
Rules used in proof :
functionEquality,
cumulativity,
lambdaEquality,
sqequalRule,
instantiate,
because_Cache,
independent_pairFormation,
independent_functionElimination,
productElimination,
hypothesis,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,s:coSet\{i:l\}.
((a \msubseteq{} s)
{}\mRightarrow{} transitive-set(s)
{}\mRightarrow{} (\mforall{}s':coSet\{i:l\}. ((a \msubseteq{} s') {}\mRightarrow{} transitive-set(s') {}\mRightarrow{} (s \msubseteq{} s')))
{}\mRightarrow{} seteq(s;cosetTC(a)))
Date html generated:
2018_07_29-AM-10_03_14
Last ObjectModification:
2018_07_18-PM-08_49_26
Theory : constructive!set!theory
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